Are there any of these tiny computers that will run a significantly long time (say 24+ hours continuously) on a pair of AA batteries?

I think you'd need to buy/make your own power adapter to power it with AA batteries.

It might be better to buy/use one of those cell phone recharger battery things to power the device via USB. Considering those things can recharge a phone a few times on a single charge, I think it should power an SBC for a fairly significant time.

The Onion Omega will run on two AA batteries, it's operating voltage is 3V3.  I have an Omega with the LiPo battery dock, haven't done anything with it, but I could charge it up and see how long it runs for.

From here:

Expect a 1500mAh battery to last about 10 hours, in some cases, up to 12 hours.

Can always use a bigger capacity LiPo or parallel a few cells, grab a 4000mAh phone battery should give you enough power for a day.

I updated my post but I can't seem to find the Power Dock in their store.

Still shows as available al-la-carte under the Omega2 Kickstarter campaign.  You can still order the Omega2 (Plus) + Dock via the campaign by the looks.