A travel poncho. You can invest in a more expensive poncho, but all I use is the compact, plastic kind (a buck or two at Target, Wal-Mart, etc.). Better yet, bring two or three, since they're awkward to fold down again once you've used them. They're not particularly fashion-forward, but when the rain's pouring and you've got a ways to walk with your backpack on, they're priceless.
Plastic garbage bags. I always bring several of the white, drawstring kind, and a few of the larger black kind. They're great to wrap around your backpack or luggage in a pinch, or to wrap your muddy sneakers in if you have to pack them with the rest of your stuff. They also work as great laundry bags.
Ziploc bags. I never run out of uses for them, rainy season or dry season.
A waterproof bag. Crucial for your camera/iPod/etc. I have some of the heavy-duty kind guaranteed to be waterproof, which are great for boats, and even better for boats in the rain. But typically, I use a clear, zip-up makeup case (available at most drugstores) for my camera when I'm outside in rainy season. They're easier to open when confronted with an amazing shot!
Waterproof shoes. Easier said than done, I know -- I haven't found a pair that's totally waterproof yet. However, rainy season in Central America is frequently warm enough for flip flops or Tevas. Since my feet are going to get wet anyway, I often opt for those.
A windbreaker, water-resistant hoodie, or jacket. Rainy season can also be chilly, especially if you're traveling at a higher altitude. Make sure you bring something to keep you warm. The less easily rain soaks through, the better -- but there's no shame in donning a poncho once it starts to pour!
An LED flashlight. You should bring one of these anyway, but rainy season often means cloudy, star-free nights. LED flashlights are small, bright and long-lasting.
A towel. You'll thank me for this one. Some travelers bring chamois/ShamWow type travel towels, but I prefer to bring a thin, inexpensive beach towel I can roll up tight in my backpack. That way, it does double duty on sunny days, too.
An umbrella. For obvious reasons! Travel umbrellas (or any small, portable umbrella) are best.
Pants other than jeans. Because wet jeans are THE WORST.
Last, and most importantly: a good attitude. Rainy season in Central America is unpredictable by nature, with storms rolling in when you least expect them -- sometimes right in the middle of your long-awaited plans. You have to be prepared to let those incidents glance off you. If you ask me, the bright green countryside, the lack of tourists and the cheaper prices more than make up for scattered rainshowers. And honestly? Some of my best Central America travel memories take place holed up in a bar or restaurant, sipping a tropical drink, and watching as a storm slams into the sea.