Talk to the guys at Silver Bow Fly Shop in Spokane and they'll steer you right.
Other than the outlet flow of Rock Lake, the Spokane River is the closest river/stream with the Little Spokane river right there also and then across into Idaho, you'll find the North Fk as mentioned plus the St Joe (top spot) just south of that, or slightly bigger waters just over across Idaho into Montana. About the same distance into Montana but a bit north is the Kootenay River out of Libby. Heading back west from Rosalia you'll also find a few streams once you cross the Columbia and head up the Cascades. Spokane/Rosalia area has many great trout lakes and is also well known for it's great stillwater fishing for trout as well as lake fishing for many other species and Sprague Lake is west of and right near your cousin and is a hot spot for fly fishing the flats for huge Carp! For a quick day trip drop over to the outlet of Rock Lake for some bruiser Brown Trout, or head out further yet and Banks Lake is a Carp anglers dream plus some other nice fishing also.
Up out of Moses Lake, west of Rosalia, there is a primo Spring Creek fishery called Rocky Ford and here's a bit on it: Rocky Ford & Desert Creeks
The folks at the Worley Bugger are a superb source of info also especially along the Cascades and into the Columbia Basin. Silver Bow gets the edge for up to date Spokane area info: Silver Bow Fly Shop, Spokane Wa - Making Fish Nervous Since 1988
Just south of Rosalia, you can find some neat streams but they are a bit further than the Spokane options. The Grande Ronde, and the the Tucannon come to mind as tribs to the Snake and all within a couple hours mol.
Good luck and have a great trip!