Are there any industry standards for setting the field of view (FOV) for FPS games or is it solely based on trial and error? Are there any guidelines for choosing the correct value?
There's no generic solution, as it essentially depends on how far away from the screen you are as well as the dimensions of your screen.
Imagine the screen being a transparent window: Depending on how far away you're sitting (and the screen size of course), you'd be able to see a different amount of the world behind you, with anything outside a specific angle being covered by the window (screen) borders and the wall (or outside world).
If you'd like to measure this for you personally, take your head's position and imagine two lines (or planes) from your head to each of the screen's borders. Measure the angle between those and you've got your own personal preference for a good FOV.
This is especially important for FPS games. It's less about immersion, but more about being able to play the game without getting motion sickness. Not everyone is similarly affected and some might have no problems with game x, while having issues with game y.
In general, just play safe and don't force any fixed angle. Instead, allow the user to choose an FOV. To avoid cheating, you should still limit the angle, but I'd suggest you do so leaving much room for configuring the game to the user's likings.
Personally, I'd suggest allowing any value between 60° and 120°, which should be perfect for pretty much anyone. This can be a challenge for level designers as well as artists, but overall it should greatly improve user experience as well as review scores and the like (I always love TotalBiscuit going to options first, hunting for a FOV setting, and then starting to rage if it's missing (which I consider a completely legit and understandable reaction)).
60 to 90 degrees is probably the standard range. As others have mentioned, it's important to make this adjustable so players can specialize it to their own setups.
It's important to consider that players have monitors of different aspect ratios. You'll need to think about how you want the FOV setting to be applied: as the vertical FOV, or as the horizontal FOV. Diagonal FOV is also a possibility. In any case, the exact vertical and horizontal FOVs will vary with the monitor aspect ratio as well as with the overall FOV setting.
Another thing to think about is multi-monitor support. Both NVIDIA and AMD's multi-monitor gaming systems work by causing a group of monitors to appear as a single giant screen to the game, in which case the game needs to render with a super-wide FOV. Ideally, the game would detect this and automatically make the necessary adjustments. This affects HUDs, menus, etc. as well - you probably want them to still appear on the primary monitor when using a multi-monitor setup.