All of Jackal's ideas are great.
My concern with Nome would be that's it's still pretty early (depending on the year) and while the roads out into the bush on the Seward Peninsula are great, you'd likely be carried off by mosquitoes when you get out of the vehicle.
Don't know if you're using cash or miles, but I'd pick Kotzebue, if for no other reason than you can stick a toe in the Arctic Ocean, and as it's above the arctic circle, you could experience true 24h sun in June. If you want to do a "whistle stop" tour, then the same plane that flies to Nome stops in Kotzebue (or v.v.) so you could hit both in a quick tour - Bering Sea and Arctic Ocean in a one.
The other option, pretty much the opposite of the above, would be to head back to SE Alaska and visit Sitka. It's off the main cruise ship routes due to docking limitations, but it's a lovely and historic town, set in fabulous scenery, with lots of cool things (e.g. Fortress of the Bear) to occupy your hours. Easy plane or ferry access from Juneau.
If you pick a return to SE Alaska, see if you can take one of the AS "milk runs," AS 66 or AS 64; for example AS 66 stops in Cordova and Yakutat before Juneau (and continuing to Seattle.) AS 64 hits Juneau, Petersburg, Wrangell and Ketchikan before SEA. If you like fun landings, these are fab.