OldManNoob:
But what if the card is corrupted (likely?) or lost in the pocket of the previous user, or simply not the right cards for any of the current need?
Part of the appeal of RFID is that it's so cheap that tags can be disposable (especially the very simple ones that are basically an electronic bar code), so that should not be too much of a problem. Another simple way of making sure you get back the cards is to have the players pay a deposit of a convenient amount (as in a single smallish note of your currency).
If you have a mostly fixed and limited set of timings, then indeed it makes sense to hard code them on the Arduino. In that case you can probably use the electronic bar code idea: that contains a number big enough to store player ID and timing set and maybe some other info.
I assumed it'd be more of a case by case basis - then you'd probably need a bit more complex an RFID with sufficient memory for your timing (but with some smarts, you can store lots in information in just 32 or 64 bytes - especially if you go bit-level rather than byte-level you can gain lots of efficiency: store start and stop time in a single byte, for example). In that case you also need a more complex system (a laptop maybe) where you can conveniently enter the custom timing table and store it on the card using a regular programmer.