First, always work backwards!
Find her death cert, it will usually list the parents' names if known.
Try to find out who adopted her/what agency was used.
If all else fails, try to contact the LDS in Utah - they keep records or everyone there and might be able to do a lookup for you. Try to remember as many names as you can, and if you can even narrow it down to a specific year (of birth/adoption/marriage/death) that makes their searches easier.
Try to find house deeds, marriage licenses, car registrations, anything with the mother's name on it, somewhere you *should* be able to figure out her maiden name - whether the one she was born with or the one her adoptive parents gave her.
Try the main library for the state in which she was (or was most likely) born if you can, some are searchable online.
If there's any way to figure out/if you know the foster agency surrounding your father, they may be able to give you information surrounding his birth records. If not, they can probably point you in the right direction.
Brick walls are what get most folks 'addicted' to genealogy

Hope I've helped you some! I have links for OH and KY if you need any later on (all contain info older than 1960's).