The name 'Bell' appears to be an abbreviation, but if so Bell's full name remains a mystery. Given the Hobbits' common practice of giving flower names to their girls, it is perhaps short for 'Bluebell' or something similar. She may even have been named for Bilbo's mother Belladonna, who would have died at about the time that Bell was born.

The evidence here is rather circumstantial. We're told that Hamfast and Samwise lived together in Bagshot Row at the time of the Party ('Both father and son ... lived on the Hill itself, in Number 3 Bagshot Row' - The Fellowship of the Ring I 1). The wording here seems to imply that they were the only occupants of Number 3, but certainly does not state that as a categorical fact. In any event, we can be sure that Bell lived at least until the year III 2983 (1383 by the Shire-reckoning) when her youngest daughter Marigold was born.