I haven't been to the Brooks, but have been to several other rivers in the area. My understanding is that the Brooks isn't that big a river. I have almost always used dry lines for rainbows, usually with egg patterns. I use about a 9 ft leader, and split shot ahead of the fly. However, you will be there before the egg laying is happening, so you'll most likely want to be ready to fish egg sucking leeches, sculpins, smolt imitations, and standard dry flies. I have done well on mice on other rivers, but, again, always later in the year. I'd plan on a sink tip line and streamer fishing, and maybe you'll get lucky and they'll look up.
If you tie flies, Alaska can be a great excuse to tie up some wild creations, because the fish can be pretty aggressive. I have a pattern that has a purple tail, a ball of orange chenille at the butt, purple chenille body, black plamered hackle, white rubber legs, and a gold tungsten cone head. That would be your one eyed egg sh*tting, rubber legged, cone headed leech.
For the sockeyes, I use nymphing techniques with small sparse steelhead like flies. You can use either a floater or sinker line. The key is get the fly down to the fish.
You should get Tony Route's book, Flyfishing Alaska. Also, check out http://www.akflyfishers.org/
Under the news and articles section is a story I wrote about fishing the Alagnak, which is very close to the Brooks. See also my site at www.oz.net/~andrewb/home2001.htm
for another story on fishing up there. Also some pictures of some minor fish.
If you like, I've got a spreadsheet with a gear list that I could send you that has a list of tackle that works well. E-mail me at firstname.lastname@example.org if you want a copy. Hope this helps.