How can I prevent my ice cream bucket from drying out?
Well, if it was a boat, you'd swab it every few days ("swab the desk") so it stayed wet and swollen.

You could also treat the bucket with a food-grade mineral oil.  You can find it in most kitchenware stores for protecting cutting boards.  The oil will evaporate much slower than water will, but it will still evaporate.

You might try putting the whole thing in a large plastic bag or similar container, so whatever moisture evaporates stays in the air surrounding the bucket, slowing further moisture loss ... but moist, stagnant air isn't good for long periods of time, either, as you could end up with mold that would be even more annoying to deal with.

Or, as the bucket's only touching the ice &amp; salt mix, and not coming in direct contact with food, you could always use wood treatments available from any hardware store. 

update : I was reading up on walnut oil, and wikipedia linked to a woodworker who recommended a walnut oil &amp; beeswax finish (1:2 blend) for finishing wood that would come in contact w/ food, but that you should re-coat each year.  I assume the beeswax would give it longer durability as compared to straight mineral oil sold for cutting boards.