What to do if a player expected much more than DM gave?

The Right Time for Roleplaying

The accepted answer is wonderful; GM's should absolutely ask a player to state their intention when it's unclear what is expected. I would add one more thing I noticed when reading your question:

but my friend decided to intimidate them, roleplayed and rolled a natural 20.

I've had great results in managing player expectations by asking players to roleplay an action only after the GM has told them what the consequences might be if they succeed or fail.

Whether that roleplaying happens before or after a roll: a player knowing what the consequences are can incorporate those outcomes into their player fantasy before they spend too much time developing that fantasy, thereby preventing a disconnect between the player's roleplaying and the reality of the dice roll.

Turnabout is fair play

All of the above answers have good discussion about setting expectations, both moving forward and trying to help the resentful player understand the past situation.

I would just suggest one more thing for trying to help the player contextualize - what if it had been they being Intimidated rather than the goblins?

That is, suppose the goblins had tried to use Persuasion (or Deception or Intimidation) on their character to convince them to betray the party and join the goblins. Would a natural 20 on the side of the goblins have been sufficient to deprive the character of autonomy or acting according to their nature? Would the DM have been justified in saying, "they rolled a 20, you have to do what they say"? In trying to get the player to understand the difference between a natural 20 meaning "you get exactly what you want" and "you do the best you possibly can given the situation", it may help them to realize they may not want to play under the first conditions if NPC's and monsters have the ability to use the same tactics on them.

Perspective

While there are already many fine answers to this question, they all (currently) give only abstract advice for talking to the player: "Come to an agreement on natural 20s," "Talk about expectations," etc.

These are good answers, with good advice, so no disrespect is expressed or implied. But they're not concrete. My contribution is two specific questions to ask the player:

1. "If this broke down into combat, how many natural 20s on attack rolls do you think it would take to win the fight?"

2. "Then why do you think one natural 20 on an intimidate check is going to do it for you? And absolve you of all the hit points, spells, and other expended resources this would require?"

Hopefully the answer to the first question is more than one, or this obviously won't work, but at least you'll understand the true depth of the problem.

And honestly, I've sort of palmed a card with that argument, coming close to the claim that a skill check success should have roughly the same effect as an attack roll success. And that's not true, either. The real precise argument I'm aiming for is not equivalence, but "Surely one of these (skill check) should not be equal to four or five of those (attack rolls)!"

Once that perspective is (hopefully) imparted, then you can start to have meaningful conversations about natural 20s, and skill checks, and how much utility to get out of them, and how many you might need to accomplish a larger goal, etc.

Then you can point out things like:

• "So really, one roll having the effect of changing the price seems pretty reasonable," or,

• "Maybe next time, one of them will break ranks and run, but that's completely at my discretion," or,

• "So honestly, this has to take more than one roll. And I hate just having players make the same roll over and over again, intimidate, intimidate, intimidate. It's boring. How would you have followed that up to get another success?" or,

• "But this is still a conflict. But now it's a conflict of intimidation vs leadership-- what would have have done if the leader decided he needed to deal with you personally in order to remain leader, and challenged you to a personal fight?"

(This is in accordance with my usual strategy of converting what the players expect to be one-roll successes into longer, more varied challenges requiring multiple successes at multiple skills.)

Unfortunately, the player was upset, there is no to little you can do with that I'm afraid. However you can prevent issues like this one in the future by setting clearer expectations as the DM.

Come to an agreement on nat 20

Players love rolling natural 20, it feels like a triumph, earlier editions also had special rule for "critical success" for skill checks which could still help players when the DC was too high.

With its "bounded accuracy" principle 5e doesn't have this rule anymore, but many tables still use it, because it feels cool. In the end of the day, we all play to have fun. You might ask your players, if they want such a rule. You should also agree, what does it mean exactly — does it let anyone achieve the impossible, or just yields additional results. When all players agree on this rule, all you need to do is just follow the agreement.

Set the expectations beforehand

Player agency can be defined as an ability to make meaningful decisions, it also implies that consequences usually meet player's expectations (which was the problem, I guess). That doesn't mean there should be no secrets nor surprises. But players should understand what they are capable of, they also should know the cost, risk and possible consequences if things go wrong.

Develop a habit of setting expectations ("what are you trying to achieve?") before asking for an ability check, when it's not something obvious (most social things are not). If your friend knew there is no way goblins let them go for free, (s)he probably wasn't so disappointed by the result. Warn players when they are trying to do something risky.

Expectation-actuality discrepancy could also be a communication problem. In the situation described above your friend probably saw it as safe and felt themselves in control. But in reality it could be quite the opposite — the goblins probably planned to kill everyone, until a lucky intimidation attempt saves the day.

This is what your player is expecting:

This is not how 5e works (and as a matter of fact, not how most RPG systems I am aware work).

It seems your player is not familiar with the system (it's comprehensible - it's practically their first time playing), and therefore their expectations are completely off from the actual game design.

Talk about these expectations

The system provides a baseline on what the expectations from the players (DM included) should be. That doesn't mean the table can't talk about it and change it if they feel the game will be funnier and more appropriate for them. If you guys can find a middle ground on the expectations for the game, that's the best outcome.

The Rules

Again: the player should be familiar with the book and the rules. There are a few things that the player is messing up here. First, they are completely stepping on DM's territory trying to dictate how the NPCs should behave. That is DM's job. See this related question about DM and players mixing their roles and turning the gaming experience into a very annoying experience (I can assure you the experience was annoying: the question is mine). So, let us remember the basic foundation of the game, described in Page 6 of the PHB, under How to Play:

1. The DM describes the environment.
2. The players describe what they want to do.
3. The DM narrates the results of the adventurers' actions.

Period. The player influences the world through its actions, and some tables may have the players (the real people) be part of the world construction as a whole (Korvin and Nitsua may help you more with that than me), but, ultimately, in D&D 5e, the DM dictates the results of the players' actions, and other than stating their dissatisfaction, the players should not feel entitled to have anything more than that.

Furthermore, the rules are clear on how ability checks work.

To make an ability check, roll a d20 and add the relevant ability modifier. As with other d20 rolls, apply bonuses and penalties, and compare the total to the DC. If the total equals or exceeds the DC, the ability check is a success — the creature overcomes the challenge at hand. Otherwise, it's a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM.

Now, the problem here is that the DM and the player had a differing view on what a success means. The played succeeding on intimidating the goblins, which means they have succeeded. But for them, this success should imply on yet another consequence: giving away Sildar for free. This is not the action the player described they wanted to do in Step 2 from How to Play - what they described was intimidating the Goblin. If the player wanted to intimidate the goblins in giving away Sildar for free, that is the description they should have given - and then the DM should not even ask for a roll: just tell them that is impossible, the goblins are not willing to do that, no matter how high the roll is.

In particular, as mentioned in other answers, a nat 20 has no special meaning in ability checks, and, in fact, has a very limited special meaning even in attacks - it is a critical hit, but that is all. No limbs being dismembered (at least not by default - you can include that with some DMG optional rules), no instant kill, no extra anything - just extra dice being rolled in the damage. Going back to talk about expectations, you may decide to house-rule otherwise, if the table agrees, and make nat 20s have special meaning. But keep in mind that it works the other way around - NPCs also roll dice and also have nat 20s, and your players may not like when that happens. In my experience, trying to give special meaning to nat 20s and nat 1s in 5e has always resulted in frustration and has shown to be a bad rule, but that is my personal experience and your table may find otherwise.

It could have been gradual

So, after the first intimidation, the goblins decrease the price. The player (and therefore the character) is still not happy. Why stop then? Continue intimidating, continue bargaining. Roll again. They could further succeed in getting Sildar back for free, or maybe they would roll badly now and fail. Point being: if they are unsatisfied with a result, they can try to improve that result in game, not complaining about it to the DM and saying it's unfair or whatever, and the DM can guide it so the player (and the character) is motivated to try harder and get even more.

Extra comments

Lost Mine of Phandelver is designed for a group of four or five characters, not three. Keeping the encounters as they are is effectively making the campaign considerably harder than it was designed. The 6 goblins are worth 600 Adjusted Experience against 3 level 1 characters, and that is not even considering the HP buff on Yeemik. A Deadly encounter for 3 level 1 characters is 300 XP. And that is happening after the Goblin Ambush, possibly fighting against some Wolves, and the goblin scouts in the entrance of the cave, and, since you already got the frog jade statue, after defeating Klarg (the Bugbear). Thus, you are completely right that the goblins have a huge advantage in this encounter and would probably not be scared easily by only three puny-looking adventurers.

But I am saying that so your DM/GF can keep that in mind: if you guys are running the adventure with 3 players, she will have to re-balance some encounters, or the campaign will be harder than you may want.

Sometimes, Giving in is okay

I will also add this suggestion as someone who has had many problems similar as yours: Sometimes, giving in to the player expectation and allowing them to have it their way, unless it is going to completely break the game by doing that, is completely okay. In this scenario, allowing the players to get Sildar back for free is not game-breaking, and if it was going to keep the table from arguing nonstop with little reason, honestly, I would have done so. I am not saying the ruling was unfair, and I am certainly not saying the ruling was wrong. I am saying that ruling otherwise might have mitigated many problems that didn't need to exist. The player didn't behave the best way either, but it's not the player who is asking the question, it is you, in behalf of your GF, so that's how she can improve: just give in sometimes. Allow the player to feel rewarded.

It's just 40 gp worth of gold. In the next few days you will be getting waay more than that in the adventure. The players don't know it, so they may actually be worried about the gold, but the DM knows, so they shouldn't be worrying so much about it.

Sometimes, it annoys me, as the DM, to give in to something that may not make complete sense to me (like in this case, why the goblins in numerical advantage are scared enough to give up that easily? I don't know), but honestly, it annoys me way more to have to argue with a player for minutes and get people angry, frustrated or leaving the table because of that.

Natural 20 has no bearing on Ability Checks

The rules for Natural 20 state:

If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. This is called a critical hit, which is explained later in this chapter.

This is strictly for attacks. There is no such clause under Ability Checks:

To make an ability check, roll a d20 and add the relevant ability modifier. As with other d20 rolls, apply bonuses and penalties, and compare the total to the DC. If the total equals or exceeds the DC, the ability check is a success — the creature overcomes the challenge at hand. Otherwise, it's a failure, which means the character or monster makes no progress toward the objective or makes progress combined with a setback determined by the DM.

So by the rules, just because the player rolled a 20 does not mean they suddenly get their heart's desire. It just means they did the best they possibly could.

Most DMs house rule that the skill check works on a sliding scale. So based on how much you exceed or are under the required DC guides how well you did. So if you only needed a 15, but totaled 23 on the roll/score you might get more than just a pass/fail.

This is a house rule, but it's a very popular house rule. So it's likely that the player feels jilted because they got the 20 when it really mattered but didn't suddenly become master of their domain.

In this case, the DM weighed out everything--number of PCs vs goblins, bargaining chips, environment, the deal that was currently on the table, etc--and made a pretty high DC. But just because they beat the score doesn't negate everything else. The goblins still had the upper hand; with numbers, a hostage, and maybe even a hidden escape route if things went bad. They had no reason to just give up after one good roll.

However, it would not be out of the realm of possibility for that to happen after a number of successful intimidation checks. The first one just made them realize they were biting off more than they can chew. The next makes them doubt their contingency plan. And so on.

So the player just needs to understand the rules and the DM needs to explain what the rolls would accomplish.

The ruling seems very reasonable

Originally the goblins were at a reasonable advantage and knew it. Successful intimidation made it so they weren't that sure anymore, probably assuming that the PCs were more dangerous than anticipated, making them (supposedly) no easy victims. In this case, taking a more reasonable price in the negotiation seems very logical.

A natural 20 does not make the impossible possible

Your other player seems to assume that a 20 on intimidation makes any foe flee. However, a natural 20 does not make the impossible possible, a check should only be used if a positive result is possible. The possible result here was the goblins dropping their assumption of superiority, not them running away heads over heals.

From a rules perspective, natural ones and twenties are special for attack rolls (always miss or hit; PHB p. 194) but not otherwise (e.g. for skill checks). Of course a GM can apply something similar to ability checks, but RAW this does not happen.

You could more clearly communicate what the expected outcome is

When someone attempts to do something and a skill check is used you can (and in this case you probably should) decide beforehand what can happen. The player attempting the task can describe what the desired outcome is (goblins running away) and the GM can explain what they deem possible, either "yes, that would be possible on a success" or "no, the best you could do would be to make them reconsider the price". That way nobody can be wildly disappointed on a good roll.

Discuss about expectations

To avoid problems it is always useful to discuss what everyone expects from the game. You should ask if the player would be okay if they knew the best they could hope for, or if they absolutely need incredible things to happen on a natural 20. If it is the latter and the other participants cannot accept that, you might have to ask the player to leave. If the expectations are too different that is the only option. But trying to compromise first is almost always worth it.