I think it is meant to be some of the 'Tookish side' that Bilbo experienced. He's the same age as Bilbo was when he had his adventure, and he's grown up his whole life hearing about Bilbo's adventures. So, it's a natural part of his heritage. It's always been there, just, 'not yet.'
But Gandalf was also a key ingredient in getting Bilbo to run out his door without a pocket handkerchief to join the dwarvish adventure, so giving him some credit here might not be remiss.
I think Frodo's reaction surprises Gandalf a bit, so it's not a specific reaction he was trying to bring about, so much as he goes throughout the world kindling hearts all the time, and that's what happens when you kindle a hobbit! So good insight on the possible use of Narya there.