I like an uke that has volume acoustically. To get good volume, you need a quality top wood that vibrates well. Of course there are other factors like the bracing, thickness, finish, and even rigidity of the back and sides that all contribute towards the volume of the instrument.
I like an uke that has the potential to play both, loud and soft and still sound good. I think guitar guys refer to this as presence. For example, I'd like the uke to sound good when strumming hard, yet still sound balanced even when strumming very soft, that's presence.
Like others mentioned, clarity of notes picked and strummed is important. A test would be to pick every note on the fretboard and listen how clear each note is picked. Then, do a chord like a barre chord, and work up the fretboard, is the chords clear all the way up? That's clarity.
I also like separation. As you chord, do you hear the separate notes that make up the chord, or just a glob of sound. When you have separation, it allows the player to emphasize certain notes and shows off the players touch.
I also like sustain. An uke that has many of the above characteristics will also have very good sustain. For an uke to have good sustain, the top has to keep vibrating. You won't find "good" sustain in an uke that has too thick a top and too heavy bracing.
Strings tend to allow the player to tune the uke bringing out certain characteristics like volume, clarity, and sustain, but only to the degree of how well the uke is built.
To add to the discussion on sound, it's also important that the uke be durable. An uke that does not hold up well, will not sound well. A thin, minimally braced top may sound great initially, but in time, it may bow or even pull off and separate. I've seen this in even the most popular ukuleles. When the top starts bowing, the scale length is affected and not only do you have an uke that looks pregnant, you got intonation problems.
Finish also adds to the durability of an instrument, however too much finish could possibly restrict the top movement, and therefore affect sound. Some will say, Nitro will kill the sound and the only good sounding uke or guitar is one with hand rubbed oil. Others say that the experienced builder is able to build a very good sounding instrument, whether it be uke or guitar, even with nitro as the finish. I tend to believe the later statement. I like a nitro finish, it protects the instrument, it brings out the richness in the wood, and it's easy to clean up.
A good way to experience the differences in sound, is to play and compare many instruments.
Like Aaron, I feel topics like this can be very educational but I'm sure he's been "in the closet" because the information shared is not always well accepted. With more players having had the experience of playing a well made instrument, perhaps there will be constructive discussion. My best advice to those who haven't had a chance to even play an instrument like a Kamaka, Kanilea, or Koaloha, try it. But don't stop there, try a King, a Glyph, Moore Bettah, Kawika, Howlett, Aaron Oya, etc. To me, playing ukes like this is like traveling the world, it will expand your thinking and give you a better appreciation of this little instrument.
Whew, been holding that in for awhile