I'd bring bedding, including any air mattress, snacks, possibly whole meals, drinks, games, books, medications, toilet paper, paper towels, etc. It's essentially indoor camping. I wouldn't bring anything too valuable if possible. Security can be spotty at shelters and in the parking lots, especially if people are freaking out.
Boredom is probably the biggest problem I've seen in shelters. Most shelters don't fill up or explode into anarchy, but instead linger on for a week or three with a dozen families or less in them. The kids especially get bored quickly.
Indoor camping I can relate to, I've done a lot of backpacking, including staging out of school auditoriums and other "mildly mitigated bloody great room" locations.
So it looks like my priorities should be creature comforts for sleeping (mask, earplugs, bedding), amusement (books, games, etc), and hygiene/otc medications.
What's the best way to deal with security? I'm less concerned with what insurance will replace and more concerned with documents, photos, and other intangibles that might go for a walkie. I know they're not obvious theft targets, but I've had some less obvious stuff consistantly stolen (e.g. flashlights whenever my car gets broken into).