I don't know what your mum is like. But my guess is, if I was introducing a parent to anime, I would stop thinking like a typical anime fan i.e. a teenage fanboy. Think seinen/josei with intelligent plots and possibly innovative animation; avoid as much as possible shows that take for granted an audience that's used to panty shots, little girls/adult men pairings, and gratuitous non-sequitur. Avoid also otaku shows - she won't get it. A useful guide would be to remember Hollywood movies/TV shows she likes and try to find similar things. If she liked Cowboy Bebop:
Ghost in the Shell
Monster
Psycho-pass
Legend of the Galactic Heroes (people like Game of Thrones, why not a political epic set in space of equal/greater complexity)
Seirei no Moribito
Planetes
ANYTHING Satoshi Kon (Paranoia Agent, Millennium Actress, Perfect Blue etc... if they like this, move them onto more 'rad' Masaaki Yuasa works e.g. Mind Game, Kaiba, Tatami Galaxy)
ANYTHING Mamoru Hosoda (if she likes this, then go Shinkai, working your way from more recent works to his earlier works, which are less sophisticated)
Deat Note/Kaiji - only if they like cliff-hanger TV shows like 24 or Lost
If your dad likes mafia, consider Gungrave over Baccano. Like, that's a proper mafia tale. Also Black Lagoon might work.
I am dying for the day when I can introduce an adult of roughly my age or slightly older to anime. I would show them the delights of animation that is mature and beautiful without needing to be depraved. Knowing the kind of person I'd hang around with, I'd go for the jugular with GiTS and LOTGH and Satoshi Kon works. Just gonna hang in there until my 'adult wannabe anime fan' arrives.