1. Get Fishping. It will replace the default, barely-audible fishing bite sound with a loud ping you're sure to hear. You'll need to have in-game sound on, though, and it's installed in a special way that is not quite the same as other addons, so make sure you read the instructions. This one won't directly relieve your boredom, but it will allow you to, say, look away from the screen for a few seconds without losing your cast, and enable a lot of the time wasters below.
2. Load up your RSS reader or your favorite website. If your computer can handle a few quick alt-tabs, your new louder ping sound should allow you to hear and tab back while you check your favorite blogs, forums, or webcomics. Make sure you allow sound when WoW in is the background under the sound options in game so the sound comes through.
3. Read a book or a magazine. This is sort of a hold over for me from the old days of Everquest, when you generally had a lot of downtime due to the fact that there was no way to regenerate health and mana for many classes besides sitting and waiting, often for at least a couple minutes. I always kept a book at my computer then, and I've picked that habit up again. If you have a stack of magazines waiting to be read like I often do, you might as well get caught up on some fishing while you catch up on your reading, right?
4. Misery loves company. No, but seriously, make an outing of it. I've personally had a lot of fun bringing 5 or 6 people along to do some fishing. You stand around, casting and chatting, maybe downing a proper brew (which will be even easier to do come patch 2.4 thanks to Captain Rumsey's Lager), and fishing doesn't seem to be quite as tedious.
5. Chat it up. Okay, so your guildmates or friends aren't as crazy as you and refuse to go fishing? It doesn't mean you can't gab their ear off anyway - I mean, if they're tolerant of that. I'm lucky enough to have a pretty active guild, so there's usually a conversation going on anyway. But sometimes it's cool to strike one up, or join, say General chat and see what's going on. I'm not saying you should take over an official channel and use it for what it's not meant for or anything, but if you can find a good conversation, either on a WoW chat channel or an outside chat channel, it helps pass the time.
6. Try a new fishing spot. If you've been sitting at the same spot fishing for a while, and you don't need a specific type of fish from that spot, see what a change of scenery will do for you. You can gain skill-ups from any body of water you can cast in, so don't be afraid to travel and see what's out there. Try a different zone, or just a different side of a zone you've already been fishing in. Make the whole thing an adventure. If you're Alliance, try fishing at Stonebull Lake in Mulgore for a while, or maybe even see if you can sneak into Ogrimmar to catch Old Crafty. The Horde can try sneaking into Ironforge to catch his counterpart, Old Ironjaw.
Really, I've found that once you get into it, WoW fishing isn't that bad. It requires a bit of "making your own fun," so to speak, but as an RPer, I'm sort of used to that. So, what about you? Do these methods work for you? Do you have another method for getting fun out of fishing, or at least staving off boredom, that you'd like to share?