.30-30 lever action. It is the gun of the west and I would also bet that many many lontime resident's of Washington hunted deer with this gun in their youth.
GoGo,
If you would allow me to be just a bit more historically correct, the Model 1894 Winchester was originally only offered in 38-55 and 32-40, thus making the rifle with those calibers the true "gun of the west". However, as you state, the 30-30 has probably taken more deer over the last hundred years than any other and is a necked down 38-55/375 Winchester.
http://www.rifleshootermag.com/ammunition/rs_3855Winchester_200811/index.html
So it depends on whether the State gun should be the gun that was used after we became a state or include the Territorial years which would be the 1873, and, like 1245A would be my vote.
Starting from the bottom: Genuine Model 1894 Winchester in 38-55, Navy Arms "Winchester" 1873 in 44-40, Marlin Model 375 in 375 Winchester, Marlin 1894 CL Classic in 32-20, and a Rossi replica of an old Overland double hammered Coach gun.
Of course, these all belong to my wifes 5th cousin once removed somewhere.