How did Tony and Cap know about the place in 1970?

I think it might hint at "The Secret Origin of Tony Stark" storyline. Howard said Maria was expecting and from his depiction, it seemed that Maria was going to give birth soon. However, Tony wouldn't be born until almost 2 months later on May 29. If you rewatch the scene, you can see how confused Tony looks and he even asks for clarification, asking "how far along is she?"

(Perhaps this should be merged with Paul D. Waite's answer, but I don't know how to do that.)

The way the trip to 1970 is introduced as Tony's idea in the film definitely makes it a little bit mysterious. The importance of the situation is emphasized by Scott Lang totally losing his cool and ranting about how the mission has failed, accusing Tony of not wanting it to succeed, and otherwise being of no help whatsoever.

Tony doesn't have an exact date or details, yet he seems completely certain that the two things they need (the Tesseract and Pym particles) can be found there. Steve Rogers wonders how he can be so sure, and the audience is invited to wonder the same thing.

TONY: I got it. There's another way. To retake the Tesseract and acquire new particles. A little stroll down memory lane. Military installation, Garden State.

STEVE: When were they both there?

TONY: They were there at a...I've a vaguely exact idea.

STEVE: How vague?

SCOTT: What are you talking about? Where are we going?

TONY: I know for a fact they were there...

SCOTT: Who's they? What are we doing?

TONY: And I know how I know.

I think the mystery of Tony's "vague" yet "exact" knowledge is answered when Tony meets his father in 1970 and Howard Stark mentions that his wife is expecting a child-- this is shortly before Tony Stark's birth.

Our memories of events from decades ago are cloudy. It's unlikely that Tony could have remembered when SHIELD was running experiments on the Tesseract or which times Dr. Pym was on site-- except that he had the one sure frame of reference of his birth date. It explains how he could be so certain without being able to give any details. Our memories tend to be very self-focused. (Perhaps even more so in the case of Tony Stark.) For example, I couldn't tell you what my grandfather's job title was, but I could tell you the exact date of a worker's strike at his company because it happened on my birthday and he told me about how he was afraid he was going to be stuck there and miss the birth of his grandson.

I think it's an effective storytelling move because it brings the focus away from the cosmic and fantastical and puts it in the realm of the personal and mundane. How do you defeat the big purple guy named after Death? With the simple connections with your family and past that make up this thing we call Life.

In the first Avengers movie, Tony hacks into SHIELD's computers while onboard the helicarrier by placing a device of some sort on the command terminals used by Nick Fury on the bridge.

Later when Tony, Bruce, and Steve are in the lab working on a way to locate the Tesseract, Tony tells Steve that Jarvis is currently hacking SHIELD's computers and, "In a few hours, I will know every dirty secret SHIELD has tried to hide."

It is safe to assume that this information would have included data on both the history of the Tesseract and Dr. Pym.

It is reasonable to assume that before starting the mission to travel back in time that Tony, Steve, and the rest of the team would have reviewed all of this information looking for places in time where they could locate the infinity stones.

This piece of information, in particular, might have piqued their interest and drawn their attention as it involved; one of the items they were looking for, a place Steve was very familiar with, a place Tony's dad worked, and the Pym particles they were using to execute the time travel plan.

It might even have been on the list of possible times and places to get the space stone but was ruled out as there was a better opportunity to get three stones at one time during the battle of New York.

The fact that Scott Lang doesn't seem to know about his alternative time and location s while Tony and Steve do, is a brilliant use of the character driving plot. Tony and Steve are both detailed oriented and would have read the briefings in-depth and committed them to memory. Scott on the other hand, well he's been established as a bit of a slacker and probably would not have read all of the briefing data.