Train one to wear a hat.
If I really needed to tell two mice apart, I’d tag the ear of one of them.
Off topic, but there was an experiment within the last year or so studying spinal healing and regeneration in rats. Turns out that a certain cheap and plentiful blue food dye is a very effective anti-inflammatory and crosses the blood-brain barrier. The only negative side effect? It turned the rats’ faces, paws, and tails dark blue.
Of course, not everyone might consider that negative.
I got two of my nieces two white mice a few years ago. The kids named them, but telling the mice apart was a problem.
So I experimented with food coloring and Sharpie pens.
I found that food coloring gets licked off within a day or so. Plus it doesn’t apply too easily. I ended up with purple and red fingers long after the mice were white again. The mice looked pretty cool while it lasted though. 
Sharpie stripes were easier to apply but still only lasted a day or so.
The way I solved the problem of telling them apart was to note that ones tail tip was blunted. Then I pointed that out to the kids.
Never did figure out how to dye them.
Landing strip or full Brazilian? 
Blue Mouse Group
Because it’s fun to be able to relate to the individual pets, even if it’s only for your own sake. Most cats don’t really have a use for names either.
And sometimes it’s practical to know so you don’t end up giving the same pitch black gerbil the same treat 3 times, and nothing for his two brothers. Grr…
Wrap one in duct tape.
It will be the gray one, and most likely where you left it.
[Scrooged]
Did you try staples?
[\Scrooged]
Tattoos?
Instead of dying the fur, where it easily comes off, why not dye the tails? The tails are mostly skin, right? Judging by the comments above it takes the food dyes longer to come off the skin than it does the fur.
Or start feeding one of them blue food. It could be an experiment like the celery between 2 glasses of dyed water.
Food coloring works, and it lasts for a couple of weeks.
You’re a Bad Man. (And I’m bad for laughing so hard at this.)
[Jimmie Walker]
Dye-no-mice!
[/Jimmy Walker]
:smack:
It’s not unheard of!
Ferrets from certain breeders are customarily given a little tattooed spot or two in the ear, to confirm that animal went through the spay/neuter and descenting operations.
I’d be reluctant to go to those extremes on my own pet, though.
If you’re going full Brazilian, you might have been better off just getting a naked mole rat. 
(Both of my cats have tattoos in their ears for identification.)
Implant a rice-sized ID chip like they have for dogs and scan them whenever you see them.
Or just give one of the several subdermal implants. You can call it the knobly one. Or grow an ear one one and a nose on the other!
Traditional methods for telling animals apart include branding and ear notching. I’m not sure how one would go about branding a mouse. At least, not if you want the mouse to survive. Most branding irons are BIGGER than a mouse’s body, so you’d have to heat up some small metal object.
Ear notching might actually be a workable permanent solution. However, I’d probably just go with clipping some fur periodically. You know those little personal shavers? Get one for the mice.
Seriously - I think everyone is over-thinking this. I used to keep mice, and would put a few drops of food coloring on their backs to tell them apart. Like I said, it lasts a few weeks, which is a significant percentage of the mouse’s lifetime.
This is the SDMB - it’s what we do.