Chapter 12:
Your father left this in my possession before he died.
It is time it was returned it to you. Use it well.
A Very Merry Christmas to you.

Chapter 17:
"Your father happened to leave it in my possession,
and I thought you might like it." Dumbledore's eyes twinkled.
"Useful things... Your father used it mainly for sneaking
off to the kitchens to steal food while he was here."


I think JKR is just yanking our chains again. I see no mystery.
Harry leaves his cloak all over the place -- on the astronomy tower,
behind the one-eyed witch, in the grass near the whomping willow...
The idea that James could leave it with Dumbledore is unremarkable.