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Bilbo Baggins always remained Mr. Bilbo Baggins. He never changed his name.
Bilbo Baggins
Frodo Baggins has the same birthday as Bilbo.
Bilbo Baggins' mother was called Belladonna Took.
"Bilbo Baggins" is the name of Bilbo Baggins. Unless you're asking for his untranslated Westron name, which would be "Bilba Labingi".