Gandalf had been to the Shire before, he seems to have known Bilbo’s mother Belladonna, a Took, he recognized that Bilbo had a strong “Tookish” side to him (Tooks being mainly of Fallohide Hobbit Stock and more adventurous with a reputation for unusual behaviour).
He also found Bilbo to be more inquisitive than other Hobbits.

@scuniper Thanks. I’m sure that was explained early on and I didn’t pay any attention to it. Seems like I would have finally picked up on it having recently listened to the audiobook version.

@SABOTEUR I had “The Hobbit” lying next to me, so I snuck a few peeks. You’re welcome.

damn it where’s Seek_Kolinahr

Frankly I have absolutely no idea. Read the book way too long ago, so if they mention why Gandlaf picked him, I frankly don’t remember. But in the movie, he does mention that he believes that, ultimately, it’s the small things that change and accomplish matters, and a hobbit would be perfect for that. They’re carefree beings, but with very good hearts, and I think I remember Gandlaf saying in the LotR books at one point that among all the races he knows about, Hobbits are most likely the ones with the purest hearts.
As well;
Bilbo, unlike other hobbits, had a sense of adventure as a child, which is a trait extremely rare in hobbits. Hobbits hate adventure and any kind of turmoil, and would rather stay still and lead a peaceful life. Gandlaf knew Bilbo ever since the latter was a kid, so perhaps this is a reason why he sensed that he would be good for this task; Bilbo’s adventurous tendencies, although only visible when he was little, made him stick out from most other hobbits.

Thorough explanation, @Symbeline. Thank you.

After careful consideration i’m going to conclude that it was in the script.

O…kaaaay @ucme. Thanks for that careful consideration.

You’re not welcome, although it was a plain fact.