All three would make excellent side trips. They're very different, though:

-Nome: barren tundra (feels more "exotic"), gold mining history (and some still active), a very frontier-type feel, three long roads fanning out from Nome that offer vistas onto truly wild and remote Alaska
-Kodiak: Alaska's "Emerald Isle," lots of dense forest cover, one long road to drive and a couple shorter ones, a few interesting sights to see including a rocket launch facility, good fishing options (both river and ocean), a real town with historical buildings and good eating options, and probably Alaska's best brewery with Alaska's least friendly brewmaster (Kodiak Island Brewing Co), potential delays/cancellations getting in and out if there are weather issues
-Cordova: Wet and rainforesty, several cool hikes up to glaciers, world-class fishing, one of the most epic and active calving glaciers in the world (Childs Glacier--take a boat tour out to it), a cool road to drive out to where the bridge went out and fell into the river, amazing tacos/burritos with fresh salmon and halibut at Baja Taco

Nome and Cordova can both also have weather delay/cancellation issues but not nearly as bad as Kodiak, where it's not an infrequent occurrence. That may factor into your plans--or at least if you do Kodiak (or, really, any of them) at the end of the trip, see if you can get it on the same PNR as your onward flight back home so that delays won't cause you to miss a nonrefundable connection.

Other options could be Barrow (more a novelty than an actually interesting place to see, but you can take a tour and get a bit of an inside look at how a real Native village lives), Kotzebue if you're interested in (and have the money and time to) getting out into real backcountry Alaska in Kobuk Valley and Gates of the Arctic National Parks, Dutch Harbor if you want to check out the gorgeously windswept Aleutian islands and site of Deadliest Catch (worse weather delay/cancellation issues than Kodiak, though!), Gustavus for a boat tour up into Glacier Bay National Park, or just keep your car an extra two days and drive over to Valdez to see Alaska's most stunning mountain pass (Thompson Pass).