Thanks to everyone who has suggested lights so far. Thanks apontes for the reference to the previous thread. That thread is the source of my original idea of the Zebralight. Pretty much the only other possibly suitable light suggested in that thread is the LiteFlux. That's why I started a new thread.
Photon rex doesn't look appropriate. It's weird form factor seems like it would complicate mounting. It's tiny battery must give it a short life, and the weird charging method would be a major nuisance. I'm expecting a minimum of 20 minutes of daily use.
Princeton Tec has a 4 x AAA light with 3 LEDs in the head that isn't very bright and has no hotspot to speak of. It does however run for days at a time and would be perfect for reading. It's 100% waterproof and even kid proof. It doesn't have a switch though, it is turned on by screwing down the bezel. .
How easy is this light to turn on and off? The light is intended for my 7 year old and she says the twisty lights are difficult to turn on and off. I'd like it to be operable one handed, which is also a strike against twisty lights. Some clickys have switches which are very stiff and also problematic for 7 year old fingers, so that could be a gotcha. Note that this is intended to be a permanent single use installation, so flexibility to other applications is probably not significant.
Several of the lights suggested so far look like potentially good options (the Gerber Infinity, the MiniMag LED) except that they are twisty.
Now I have to admit that the Nitecore D10 looks nice, but it's $59. If I'm going to spend that much and then still have to monkey around with bottlecaps and scotch tape...why not just get the Zebralight which appears to be the overwhelming favorite at the same price. Photon Pro was also suggested but it's $79 and doesn't even have a low mode. LiteFlux looks nice. But it is better than the zebralight for this application? Price seems to be comparable.
JetBeam appears to be $79, so that's more than the Zebralight. Arcmania MJP extreme III appears to be far beyond what I want to spend. (I couldn't really find any info on this other than someone sold one "cheap" for $130.) I might be able to persuade the wife that we should get the Zebralight...but something that costs $130? Forget it!
I experimented with scotch tape diffusers on my Fenix and Arc. I found that it made the beams a lot more even---particularly the Arc which has a very sharply defined hot spot---but they still had a definite hot spot. I guess more diffusion would make it more even. I didn't have any bottle caps available to try. I need a solution that's reasonably permanent, so scotch tape, or a paper towel won't cut it. But the bottle cap might work. As I noted above, though, it doesn't seem sensible to get a $60 light that needs fiddling with bottlecap. If you know of a $15 light that needs this sort of fiddling, that might make sense. The tendency I've observed is for the cheap lights to have one brightness setting...and it's probably too bright for reading in the dark.
One possible exception: AKoray K-106. This one appears to have three individually settable modes and it's $20. Any comments on its suitability for reading after application of some sort of diffuser? (I read in reviews on dx that it was hard to operate the switch.)
Questions for zebralight users. Is the clicky switch easy to use? What is the sequence of brightnesses? If someone shines this light in your face how annoying is it?