Jamie, the first believer in Jack Frost, and it was fate it would be him, because even if we just see glimpse, he is the child that likes everything that has to do with supernatural, spirits, and unknown, he is a firm believer in Santa, Tooth fairy, Easter Bunny, Sandman and stuff that comes with that, so it's actually not strange that Jamie begins to believe in Jack Frost and also seeing him.
From the beginning Jack seems to be drown to the boy, and gives them Snow day, and makes him do a crazy riding way for the sled... making it so that Jamie loses his tooth, and is excited about it.
Even before Jamie sees him, Jack grows an attachment of an older brother when it comes to him, and the fact is that it's Jack who saves Jamie's believes when it comes to Easter Bunny, he is the one that shows him, that yes, Jamie didn't dream up that night when Toothiana came for his tooth, along with Santa, Easter Bunny and Sandman.
But it's the fact that his mother nominated Jack at the beginning, that Jamie connects the dots, and snow in his room could mean that Jack Frost was actually there.
The relationship between those two is one of two brothers, I could call it
bromance, because they understand each other, and they have fun together, and
they help each other, in the end it's Jack's want to help Jamie that makes him
realize, that his center is fun, while for Jamie, Jack is the older brother
he wish he had, the fun kind of brother.