Gandalf already knew Bilbo's mother, the fabulous Belladonna Took, who had an adventurous streak before she married Bungo Baggins. He assumed that that streak would be present in Bilbo as well.
More specifically, Gandalf tells why he chose Bilbo in the
Unfinished Tales.
Quote:
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Somehow I had been attracted to Bilbo long before, as a child, and a young hobbit: he had not quite come of age when I had last seen him. 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. As soon as I entered the Shire I heard news of him. He was getting talked about, it seemed. Both his parents had died early for Shire-folk, at about eighty; and he had never married. He was already growing a bit queer, they said, and went off for days by himself. He could be seen talking to strangers, even Dwarves.
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth..
.'