My son had a child with his girlfriend. Since they’re not married, how do I refer to her. Is daughter-in-law proper?
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.
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.
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 the 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.
Ask her how she'd like you to refer to her. "Propriety" is not nearly as important as family relations.