Ok at first I thought water but they realised that was wrong, so I have come up with a new answer:
You're a cloud
I come from a brook.
Clouds are made of water, (well water vapour)
I'm first in a book.
Not sure, maybe something to do with being white? White page, white cloud?
I'm old but not stale.
Not sure again, but I suppose there has always been clouds so that makes them old.
From me was made hail.
Well hail comes out of clouds were they are made
I sound like an ocean.
Storm cloud (Thunder sounding like waves) or something to do with rain
I suppose clouds are always on the move
I'm not quite a kernel,
not sure, maybe the sun is the kernel and clouds merely part of the scenery?
nor am I nocturnal.
You can't see clouds at night.
Am I right?