<p>I'd like to print modifications for my bird feeder, both to patch over the hail damage from last summer and to try to deter the neighborhood squirrels.  I have an FDM printer (and experience with nylon, ABS, and PLA, though don't restrict answers to those if there's something else that's better), what kind of filament would stand up best to daily exposure to sun, rain, snow, etc?</p>

PET(G) is a strong contender. It is very strong and water-resistant, and as such is often used to make pop bottles.
PLA has a reputation for being "biodegradable" and therefore it is often discouraged to use PLA outside and/or in contact with water. However, PLA only biodegrades under very specific conditions which it won't generally be exposed to so it can be used (though, as a harder and less flexible material it is more likely to be damaged by hail).
ABS and Nylon are good choices as well. Basically, any plastic you have on hand will last for years, even in an outside application.