How can I test whether my compact camera’s autofocus is performing normally?

There's a few ways to go with this. First, you need a point of comparison, or a baseline to say what is "working" or "good enough" or "problem free".

This might be something like:

• As good as other Olympus TG-2 cameras, i.e. is it function at it's own spec?
• As good as another specific camera, i.e. is it functioning as well as its peers?
• Reliably achieves focus on a particular subject (at a particular distance + light level), i.e. is it functioning as well as I want/need?

In all cases, you need some kind of test subject

• ideally one that's very consistent and repeatable
• and well suited to the AF method (e.g. strong vertical or horizontal lines for a DSLR's AF sensor, or a high contrast black & white pattern for a contrast detection sensor)

You'll need to decide on a distance, or range of distances to test it

• Make sure the AF subject/target covers/fills the AF region at said distance
• This may require multiple targets, or quite large targets at long distances

You need to set appropriate light levels.

• You want to test in the range of lighting where the camera is expected to work (e.g. daylight)
• You may also want to test in low-light conditions (with the expectation of slower and/or reduced likelihood of achieving focus)
• Ideally, if testing multiple cameras, do so at the same time in the same lighting conditions (it can be hard to make lighting repeatable)
• Artificial lights would typically be required for consistent tests (you can use the camera's light meter also, to ensure you have approximately the same lighting, but it's averaging over the image so won't necessarily be quite the same at any given point).

If comparing against another camera, or other Olympus TG-2 copies, then you'll obviously need to get data from them as your baseline. If comparing against some expected AF performance (the third option above) then you'll need to decide up front what performance you expect.

What to measure?

As you mention, the AF speed is a major concern.

• This can be hard to measure precisely, but use a stopwatch and record every sample
• You'll want to take multiple samples to ensure it's consistently the same speed (or to measure inconsistencies)
• Make sure the focus is reset to the same focal distance each time you test it (e.g. by focusing on infinity or extreme macro between each test), otherwise you'll be measuring different focus adjustment times in different cases.
• Be aware of how long it takes for the camera motor to change the AF between infinity and its closest macro distance; this is part of the AF time. The other part of AF is the 'hunting' where it is moving to what it expects will be the right position, and re-testing focus.

With multiple samples, you can measure consistency:

• This is especially important at low light levels, because you can't expect the camera to achieve good focus 100% of the time outside its specified performance bounds.
• Take at least 5 measurements, maybe 10-20 if you can be bothered (for any given combination of light level, AF target, subject distance, camera)
• Calculate the percentage of tests it actually achieves focus
• Calculate the range or distribution of times taken (sometimes it hunts, sometimes it gets it right, so max–min or some variance/standard deviation may be useful)

Taking photos after each test will also give you a sense of accuracy:

• Is the AF target/subject actually in focus?
• Ensure the camera is not moved after focus (use a tripod if you can)
• Ensure the target was clear and directly facing the camera (e.g. a printed target should be viewed straight-on rather than at an angle)
• Take photos at the widest aperture (smallest number) possible, to minimise the depth of field for this test (it will amplify any errors in AF)
• As with the consistency above, you'll want multiple measurements to calculate a percentage of how many were "in focus"

How did it compare?

Without getting into a statistical analysis of the results, you should now be in a position to make some basic comparisons. Did the AF perform similarly to other copies of the camera? How did the camera's speed, repeatability and accuracy compare with another camera? Did it get enough photos in focus, or focus fast enough for your needs? You should expect to see some shot-to-shot variation (even with perfect lighting, focus target, etc). It may help to be able to provide this information to the manufacturer or sales person, who can compare your numbers with their own experience (e.g. "no it shouldn't take 5 seconds to focus on a person standing 2m away in daylight", or "yes, half a second and 95% accurate is very good for this type of camera, stop whining!")

Unfortunately many camera reviews are quite vague about AF speeds, and if you've come from using modern DSLRs to a point & shoot, then "fast and mostly accurate" may mean something entirely different to someone coming from a cheap point & shoot. Even with in-depth reviews of expensive DSLR lenses, I rarely see more than "the AF speed will not disappoint", or "it's a bit slow to focus". A few actually specify things like the time taken to drive the AF from end-to-end with a standard camera body, which actually lets the reader make an informed comparison.

Finally, note that camera focus depends a lot on the test target, so you may be surprised to see under very controlled conditions your camera performs quite well, even if it's not working as well as you'd like in "real life" conditions. If that's the case, then it may help to understand the specifics of your camera's AF system, like what makes a good/bad target, and tricks you can use to achieve focus (e.g. using the AF to focus on a similar, but brighter nearby subject instead, then re-framing the camera to take the shot—though admittedly many such tricks like this may be more specific to SLR with small fixed AF regions). Sometimes people also blame misfocusing on the AF mechanism, when it has simply focused on the "wrong" subject. Typically cameras will try to focus on the closest thing in the focal region, but if there's a few different items, it may either get the "wrong" one (and focus on that leaf/fence closer to you) or if the main subject is moving about or massively out of focus (it's already close to infinity) it may ignore it as 'noise' and just focus on the background (perhaps after trying and failing to achieve focus on the moving subject).