Clues in the MovieEdit
When they reach Lennox House, Baby's Stepfather can be seen filling out paperwork, checking off boxes that claim she is a threat to herself as well as others. When the camera angle shifts, a careful viewer can discern a line asking for the patient's full Christian name. Stepfather has clearly written "M." as an abbreviation of her first name, and is writing her surname, which clearly begins with an "R," followed by what is almost certainly an "E," then either another "E" or an "A," followed by three muddled letters that closely resemble "v-e-s."
This would seem to strongly suggest that her name is either "M. Reeves" or "M. Reaves." The monogram on the bedsheet pulled from Babydoll's deceased mother, "HRM," further confirms that the family's last name begins with the letter R.
But what about her first name? Babydoll is 20 years old in the movie, which means she was born sometime in the 1940s -- probably as part of the postwar Baby Boom. "Mary" was the most popular female baby name between 1940-'46, and ranked No. 2 from 1947-'49 (when it was bested each year by "Linda"). This suggests that her first name is most likely Mary.
Thus, we get a best-guess name of "Mary Reeves" or "Mary Reaves."