What is the name of Bilbo's mother?

Bilbo’s mother is Belladonna Took.

She connects Bilbo to the adventurous Took family, while his father, Bungo Baggins, represents the respectable, comfort-loving Bagginses. Bilbo inherits both sides of this family background, a contrast that helps explain his character in The Hobbit: he begins as a cautious homebody but discovers courage and resourcefulness on his journey.

Belladonna was the daughter of Gerontius “The Old” Took, the famous Thain of the Shire. Through her, Bilbo was related to many prominent Hobbits, including the Took family that later produced Peregrin “Pippin” Took. Bilbo’s Took ancestry is also why Gandalf sees some latent adventurousness in him.

A common mix-up is to confuse Belladonna Took with other women in the Baggins and Brandybuck family trees. Lobelia Sackville-Baggins is Bilbo’s relative and an unpleasant claimant to Bag End, while Primula Brandybuck is Frodo’s mother. Rosie Cotton belongs to Samwise Gamgee’s story, not Bilbo’s immediate family.