I've thought about this again. If Marie re-married the new husband could have adopted Russell/Rusty and his last name might not be White.
The Censuses end in 1930.
Do you think Russell/Rusty was born in Shoshone Co? You can contact the Idaho Bureau of Vital Records and Health Statistics.
Then you'd have an exact name, birth date and place. That doesn't help you locate him but that will help to eliminate other Russell/Rusty White's.
Maybe Marie re-married in Shoshone Co. You can contact the Shoshone County Clerk, 700 Bank Street, Wallace 83873, 208-752-1264 and request a copy of her new marriage license. I didn't find it in the index but that doesn't mean it's not there.
However, if you find her new married name, that won't help with locating Russell/Rusty. Wish I had better news for you.