I propose that aperture, the size of the hole where light is entering the camera, is the least important setting for this type of photography. Having a big opening lets lots of light in, which is great if you are only opening the shutter for a very short moment of time. However, there are some downsides to this.
First, most lenses are “soft” when you shoot them “wide open.” In normal speech, it means if you have the aperture open all the way, your images will not be as sharp. The best sharpness usually starts a couple “stops” smaller than the maximum.
Wider apertures also give less depth of field. Sometimes photographers limit the depth of field intentionally so only a small amount of the photo is in focus, but this isn’t necessarily a good idea for bird photography. For birds, it is often nice if the eye and the bill are both in focus. A very small depth of field also means that any slight error or movement and your subject may be out of that narrow in-focus band. Because of physics (optics specifically) cameras with smaller sensors, like any point and shoot camera, will have greater depth of field, so this is only really an issue for cameras with large sensors like DSLRs.
That said, one can get great photos even at the widest aperture. The loss of sharpness won’t be as much of a problem as blurriness from a photo with a shutter speed that is too slow. If I have enough light I will always try to shoot a few stops down, but this is the first thing I compromise when I don’t have enough light.
As a side note, the difference between the two images in this post is much easier to see at full size. This also means that if you are primarily taking photos to share online, the aperture you’re using probably won’t matter.