Why did Gandalf choose Bilbo?
 

If I've forgotten, please refresh my memory...But why did Gandalf choose Bilbo for the Quest of Erebor? Why did he deem Bilbo a worthy candidate?

It's elaborated on quite a bit in "The Quest of Erebor" in the Unfinished Tales. Basically the point was that Gandalf had remembered him already from before, as one being curious about the adventurous stories and wide world, and after many years overhearing him being talked about by the neighbours as a bit queer one, talking to Dwarves and strangers and, very importantly, not being attached in any way (not having a family etc).

Why did Gandalf choose Bilbo?
 

I suppose he got him at a half-off discount.

In addition we have the 'I knew the west was in for a hard time, and without leaders the hobbits would be in great peril, they had begun to forget....' a paraphrase and I'm not sure if it is from the quest for Erebor [a full version appears also in 'the annotated hobbit' and explanations as to why the variations in UT do not reveal THE version] or if it is the RotK.


But Gandalf, being 'a steward of all things' saw the need of the hobbits in general even as he saw the potential of Bilbo in particular.

To that and what's been said above, add the observation by Gandalf that Bilbo was "meant" to find the Ring. There clearly seems to be an element of destiny in Gandalf's choice, whether or not he himself was conscious of it.