I'm sure I've read of someone experimenting by just taping pieces of card with different sized holes onto an existing soundhole. Obviously you can only go smaller!
I believe these were in line with received wisdom, that smaller soundholes enhance bass response while large soundholes enhance treble.
As the soundhole "tunes" the body resonance, which I think is mainly based on body volume, changing the depth of the sides should have a similar effect (more body volume = more bass, etc). But dwizum's experiment suggests that reducing the body volume and increasing soundhole size gives more mid-range response, rather than both working together to increase treble. This suggests that body depth is a third factor - anyone able to explain how the three interact?
I think the basic concept of how this all works is as follows (please note that this comes from my knowledge of the design of ported loudspeakers, but I believe that the physics is going to be the same):
1. When you play a note on a uke, what you hear is the sum of the responses from the string, the top, and the sound hole. The string response is quiet compared to the other two, and can be neglected. There may be low-volume responses from the sides and back, but we'll also neglect them for this discussion.
2. The response from the top is directly excited by the string, and is mostly going to be a straightforward function of the fundamental tone plus harmonics.
3. The response from the sound hole is a little more complicated, conceptually. The air within the uke is going to be excited by any note played on the strings, but not every frequency will be result in an equal amount of volume coming out of the sound hole. The combination of the internal volume and the area of the sound hole will select for a specific frequency at which sound hole output will be highest.
4. If you measured and graphed the output of the sound hole across a range of frequencies, you would see that output from the sound hole is highest at the frequency selected by the combination of internal volume and sound hole area (i.e. sound hole resonance frequency). Above and below that frequency, output will drop off at a characteristic slope. As a result, frequencies far from the sound hole resonance frequency won't be strongly reinforced by the sound hole, and the sound hole output will be negligible at very high frequencies.
5. Since the total output of the uke is the sum of the top plus sound hole, you can alter the balance of sound by changing the sound hole resonance frequency. It's important to know the frequency of the lowest open string that will be on the instrument. If it's a low G and the sound hole resonates around low G or a little higher, bass notes will tend to sound boomy, or at least noticeably louder than notes played on other strings. If it's only going to be used for standard reentrant tuning, you might choose it to be a bit lower than open C, so bass from the C string is strong but not too strong.
6. Making the sound hole bigger (edit: *smaller*) than standard for that size uke will lower the sound hole resonance frequency, which at some point may seem to emphasize mid and upper frequencies. What's really happening there is that the sound hole resonance gets pushed far enough below the string frequencies that it starts to contribute significantly less to the overall volume of sound of the bass strings. So it's really less bass, not more mid and high.
7. As far as considerations of depth and body shape go, the most important thing is going to be the internal volume of the instrument. Skinnier depth and wider top is the same as deeper depth and narrower top, or longer or shorter body, if they all produce the same internal volume. There might be a difference if you had a shape that was much thinner than the norm, or with a very narrow transition from upper to lower bout, etc. Basically, if some internal part of the uke is narrow enough to have its own resonance frequency different from the sound hole resonance, that could change the bass response in an unpredictable or undesired way. Of course, differences in body shape change the size and shape of the top, which will have its own effects on the sound, how the instrument is braced, and so forth.
8. Incidentally, changing the shape of the sound hole, within reason, probably will have little impact on the sound if the area remains the same. I expect that f-holes on mandolins. violins, etc. could sound different if changed to round holes, because so much of the area of the f-hole is very narrow and long. But going from round to oval, square, or irregular, isn't going to matter much. Two smaller holes instead of one big hole probably isn't going to matter much either.