There's a baseball sized hole in the aluminum siding on the back of my house left by the previous owners that's causing some water damage on the interior wall. The houses siding is old and beat up so I don't care about making this repair look good I just want to eliminate the elements, and birds, from getting inside it. Would it be sufficient to tape over it with some white gorilla tape (my house is white) or should I really be going a more legitimate route?
Since you don't particularly care about looks, all you will need to do is create a patch out of aluminum sheet metal and attach it to the siding.
Start with a sheet of aluminum. A good source from this is from a roll of drip edge. Cut a square or rectangular piece out using tin snips, and make it slightly larger than the hole. You would want to make it large enough to cover it completely, and still have room to attach it.
Once the piece is cut, you should paint it to protect it, and have it blend in with the color of your siding. It doesn't need to be an exact match, but the closer, the better.
Attach the patch to the siding using 1/8" aluminum pop-rivets. You also should use a thin bead of caulk around the hole to keep it weather tight. Once the patch is installed, you should also apply one more coat of paint to cover up the heads of the rivets.
Tape won't be enough. Critters will get through it pretty easily.
I would cut a small patch from aluminum flashing, or preferably a spare piece of siding, and stick it on with epoxy (caulk will work too, but it won't be quite as durable as epoxy.)
Aluminum foil tape will work in a pinch. It sticks better than cloth-based duct or gorilla tape.
You can purchase trim coil although it's a bit much to pay for a small piece. Ask neighbors or relatives if they have some lying around; if they've ever used it for an addition, or replacing windows or window trim, chances are they have some left over. You can also go to a big box store and look around in the siding area, you can get aluminum meant for flashing in both coils and loose pieces, or even purchase a single piece of siding to cut up if you like.
Cut a square big enough to cover the hole and apply caulk around the hole before putting the metal over it. Drive some white gutter screws through the new piece into the siding and you're good to go.