Records depend on location -- every country and every state in the US have different laws, regulations, coverage by years, etc.
Why don't you post as much as you know (names, locations, etc.) and see what folks suggest? There are usually "experts" for locations, or record types, etc.
Off the top of my head, I'm thinking birth indexes, newspaper announcements of births or hospital stays (perhaps not if she was "illegitimate" though), and court records...
Is her father dead? If so, his estate records or will MIGHT have mentioned her (unlikely, but POSSIBLE)
Was he in school at the time? College? Alumni might know who his "girlfriend" was.
Was he working and she was a co-worker? a neighbor?
How did you find out the info? What's/Who's your source?
ANY little detail about the mother could be a clue -- occupation, age, birth place, etc. etc.