How do I refer to the unmarried mother of my grandson?

There are two separate issues here.

The first is how to express the fact that she is your son's partner. You can say "daughter-in-law" if you are not too worried about accuracy, or "my son's partner" which has the advantage of being gender-neutral and marital-status-neutral. But since she might be offended by one or the other if she got to hear that you had used it, the only safe answer is to ask her what she would like.

The second is that your son's partner, whether married or not (and indeed whether married to your son or not) is not the same as your grandson's mother. They might be the same person or they might not. So the only clear way to identify your grandson's mother is to say "my grandson's mother". Of course, you might choose to say she is your son's partner in some way and just imply that she is your grandson's mother. If you really wish to make it clear that she is both your son's partner and your grandson's mother you will have to state this explicitly.

I'm going to take a very clever observation from @michael.hor257k here, and use it to establish the proper term to introduce this woman with:

Should be daughter-in-common-law, then.

What's this word "common" here? The key to the puzzle, is what it is. From Wikipedia:

Common-law marriage, also known as ... marriage in fact, is a legal framework ... where a couple is legally considered married, without that couple having formally registered their relation as a civil or religious marriage.

The original concept of a "common-law marriage" is a marriage that is considered valid by both partners, but has not been formally recorded with a state or religious registry, or celebrated in a formal religious service.

In effect, the act of the couple representing themselves to others as being married, and organizing their relation as if they were married, acts as the evidence that they are married.

Your son doesn't need the arbitrary imprimatur of the State to know he is married, and neither do you, nor does anyone you may wish to introduce your daughter-in-law to.

The term you want is daughter-in-law. Use it in good health. And be wary of any other term anyone tries to sell you; they are fraught¹.

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¹ _{On the other hand, if you happen to be dissatisfied with the state of affairs of your son's marriage, and simultaneously wish to convey that dissatisfaction when you introduce your daughter-in-law to new people ... well, I'm not the man to help you. I wish you the best of luck in finding a term to meet your needs. Congratulate the happy couple on their child for me.}

I gather that you’re looking for a term to express your own relationship to the lady in question.

Since she has no formal familial ties to you, there isn’t a standard term for the (nonexistent) formal relationship.

Consider a similar situation: two brothers marry two women from different families. What do the women call each other? They can call each other sister-in-law informally, or husband’s sister-in-law formally, but English has no simple term to recognise the women’s formal relationship via their husbands. It is common to say that there actually is no relationship between the women via their husbands.

Likewise, you can call the lady in question your daughter-in-law informally (or possibly even formally if you take into account the de facto relationship she has with your son).

If you discount the de facto relationship, the lady has no relationship with you via your son; her relationship to you via your grandson has no simple English term (grandson’s mother doesn’t link her to you). In that case, you can take David’s suggestion to simply call her by her name.

Ask her how she'd like you to refer to her. "Propriety" is not nearly as important as family relations.

In Spanish they have simple words, for example, "Suegra" for mother-in-law, "Nuero" for son-in-law, etc. It is the same word whether or not the couple is legally married. I think it's OK to say "in=law" if the couple has lived together a few years. We shouldn't need to add the word "common" as it sounds less respectful. But I wish we had better words that didn't include "LAw" as if that determines the strength of relationships.

I realize I'm giving an opinion here, which seems to be against the guidelines, but other answers here have given opinions also. And it's factual about the words in Spanish being used to describe couples who aren't legally married, at least in my experience as a "Nuera" (daughter-in-law) both married and unmarried. .

Daughter-in-law is fine. A couple can be married “in common law”, which in many locations gives the spouses the same legal status as a formally married couple.

https://en.m.wikipedia.org/wiki/Common-law_marriage

In many of these places, the law gives marital partners rights in jointly accumated property, as well as standing in family court, probate court and so on. In fact, anyone who thinks that they’re avoiding these obligations by not getting formally married might be in for a surprise.

Common-law marriage is not identical to formal marriage, but it’s still a form of relationship recognized in law.

In ordinary conversation, there is no need to draw attention to the legal form that your son and his partner have chosen for their relationship, unless you specifically want to.

In legal matters, such as defining her in a categorical form in a will, or in an application for government services, or in international travel, you will have to be exact. In which case, “the mother of my grandson” is more accurate.

One of the advantages of formal marriage is that a certificate is all you need to prove your legal status. The common law spouse may have the same rights in theory, but in practice it may take more legal effort to enforce them.

The OP has already, within the question itself, used the least problematic term for this person: she is the girlfriend of the OP's son. That's what she was before the grandson is born, and the fact of his birth has not (in itself) changed that.

The OP is related to that person in two ways:

OP - son - the son's girlfriend,

and

OP - son - grandson - the grandson's mother.

When one refers to people in terms of the connection one has to them, one normally chooses the terms on the basis of the most direct connection. In this case, the first connection is more direct than the second one.

If one thinks that the term girlfriend fails to fully capture the nature of the son's relationship with the woman, one may use something like my son's long-term companion, my son's partner or my son's significant other. As has already been pointed out in other answers, in the contexts that have no legal implications, one may also refer to that person as my daughter-in-law: although not technically accurate, such a term may convey that one regards that person as one would typically regard one's daughter-in-law, and in some social contexts that may be more relevant than the legal technicalities.

The term my daughter-in-common-law, suggested elsewhere on this page may or may not be technically accurate, depending on the specifics of the son's relationship with the woman and the relevant laws of the particular jurisdiction. Even when it is accurate, there are very few situations in which it would be called for; in most contexts, its cumbersomeness is likely to be distracting.

Referring to that person as my grandson's mother would, as a matter of pragmatics, implicate (even though it would not logically entail) that this is the most direct connection that one has to the person. Using that term could thus lead those who hear it to think that the son and the woman have broken up. The term is, therefore, misleading if the two of them continue to live as a couple.

My son has been with the same sweet girl for 13 years. They’ve lived together for the past 8. They still haven’t gotten married, although I hope they one day will, so we just refer to her as our Daughter-At-Heart:)