Hillary had kept her own last name of Rodham when she and Bill married in1975, and went by Hillary Rodham only after her marriage. When Bill was running for governor of Arkansas, she was pressured into giving up her own name and to become "Hillary Clinton", so in order to not offend traditionalists who hate the idea of women having their own names.

I would say that her going by Hillary Rodham Clinton is a way to partially reclaim her own last name that she was compelled to give up so many years ago. Indeed, I wish she'd had the balls (ovaries?) to have resisted the pressure to take on the name of Clinton in the first place and stuck to her guns.