Thelma Gordon.
Sarah from DKR is probably Sarah Essen, Gordon's second wife and no relation to Barbara other than through marriage. However, DKR is not in continuity and Sarah Essen-Gordon was slain by the Joker in "No Man's Land."
The problem with the name of Barbara's mother is that her parentage changes constantly with all the continuity resets DC does. She's gone from being his daughter to being his niece who was adopted by Jim after her parents died to being possibly Jim's biological daughter via an affair with his sister-in-law (making her his niece as well).
Thelma Gordon is her mother in current continuity. Her father biologically is either Jim, or Roger (Jim's brother and Thelma's husband). Roger and Thelma died when Babs was 12 or 13, and Jim and his wife (also named Barbara) adopted her. Jim kept custody after the divorce, his ex getting custody of Jim Gordon Jr.