Fighting in the arena is hands down my favorite thing to do in WoW these days. I do get very bored while I'm waiting for queues to pop. Since I usually do 2v2 and 3v3 battles I'm usually looking at wait times between five and ten minutes. Drysc stated that this occurs due to a limited number of instances that can be hosted at any given time, the problem will not likely be solved in the near future.
Instead of complaining about it, I try to find something productive to do with inevitable delay. There are several in-game activities that to do with these unexciting minutes. I love to window shop the next piece of battleground or arena gear that I'm working for. It is also an excellent time to work on crafting and auctioning items. Skinners who queue up in Blade's Edge, find themselves in a perfect position for killing Bladespre Raptors for Knothide Leather.
Many players take the opportunity to hone their PvP skills by dueling outside of the major cities. It would be nice to be able to run battlegrounds while waiting, but there are good reasons for why this is disallowed. Woe would be the Warsong Gulch group that's leftover after a 5v5 team split in favor of an arena.
Outside of WoW, there are plenty of opportunities to be productive as well. The wait time is perfect for household chores or catching up on news. I find it a good time to do my reading for school. Many times I turn the volume up on my computer so that I can walk away and hear the queue pop. Sitting and staring blankly at the screen while waiting for my shot at honor and glory is relatively low on my list of things to do. I'm all about multitasking.
The queue time is a drag, but you might as well make the most of it. What do you do when you're waiting for the queue?