When did Gandalf Choose Bilbo?

[Inziladun]

Originally Posted by Morsul the Dark

So the question after that ramble when did Gandalf settle on Bilbo?

The Unfinished Tales section The Quest of Erebor addresses this question.

The scene is that of Gandalf telling Frodo, Merry, Pippin, and Gimli in Minas Tirith about the events leading to The Hobbit

"Somehow I had been attracted by Bilbo long before, as a child, and a young hobbit. He had stayed in my mind ever since with his eagerness, and his bright eyes, and his love of tales, and his questions about the wide world outside the Shire."

Gandalf goes on to say that when he entered the Shire after speaking with Thorin, he heard Bilbo was still unmarried, and was known to do odd things like talking to Dwarves.
It was then he decided to send a hobbit with Thorin and Co., and to make that hobbit Bilbo.

In another version, Gandalf is said to have wanted a hobbit with 'a dash of the Took', and a 'stolider' family like Baggins. That also led to Bilbo.