I don't know the regulations either but how about steping up to a .243 or depending on his size a .25-06.  Both are light recoiling rifles and both should be leagle for elk no matter what state you are in. 

I just did a quick search and it has to be a minimum .24 caliber for big game in Washington. 

.243 or 25.06 would be a great caliber for him and deer.

I'd go with a 243, my 9yr old shoots one now and has no prob with it.

Maybe the 243 or 25 WSSM. I have a Win Stealth 2 in 25wssm that's a joy to shoot.  It has good weight to it to tame recoil. It actually weighted too much when I bought it b/c of the bull barrel. I cut the barrel down to reduce weight and give it better balance.  You could always buy a 270/308 and put a threaded muzzle break on it. When he grows into the rifle, you can srew the break off.  My gunsmith has the machinery to make a threaded adaptor to screw on in place of a break. Check around. Good luck to you and your son.  Godspeed

Thanks for all the help everyone.  I guess my next question is does anyone have a 270 or the like for sale?  I need to get one by mid September to get him some time behind the sights.

I wouldn't overlook the 7mm-08 for a youngster. It is a neckdowned 308 and thus is light recoiling but still has enough juice to take elk ethically. A 22-250 has no business in the elk woods in my opinion. Congrats on getting a youngster out there!