It's hard to say what value pot would work best, particularly since we aren't sure how that thing is wired up, it's not as easy as calculating current for a single LED and making a good judgement. WE don't know what kind of limiting is already built in, etc..
Basically LEDs (single ones at least) can run with normal currents between 5mA and 20mA depending on the LED, so on that thought, with a 12 volt source, 20mA and 10mA sort of target currents, you'd want maybe no more than a 2K pot to be sure you can limit all the way down to a few mA from 12 volts.
But in reality there's probably current limiting built into that array, in which case you'd only need to really add several hundred ohms to the 12volt line, you'd have to experiment.
If you want the best suggestion to get you the most results from a single purchase rather than building up an experimenter's junk box, I haven't been shopping in a while so I'm not sure what standard pot values you can easily get but try for something like several hundred ohms, and if you can't get one that low, try 1K or 2K but no more, you'll not get much range out of it because as someone said most of the adjustment span will be too-much limiting and you'll have an OFF led that suddenly goes ON near the end of the dial...no real tweaking....
Then what you do with that pot of a few hundred ohms, also get a FIXED resistor if necessary and put in series along wtih the pot, and that way by having most of the resistance fixed, and a small amount of adjustment on the pot, the pot becomes a more fine tuning element and you get more control for the full span of the dial with that small pot value.
The fixed resistor represents what WOULD be the unusable range of the single pot method where most of the dial is an OFF led, and the small pot expands the small usable region of the single-large-pot over a full dial now. It's like zooming into the usable region of the pot.
So if you can find a pot a few hundred ohms, buy some fixed resistors around 100 ohms and you can connect them in series or parallel to get a usable fixed resistance that lets the small pot work well. Two 100 ohms in parallel give you 50 ohms in case the 100 was too much the first try, etc.
Not knowing how the thing works, and knowing you have 12volts, just to be safe when experimenting I'd probably put 3 of the 100 ohm fixed resistors in series, along with the pot with its dial at center (in case you guess wrong about which end is 0ohms and full rating, better to have some resistance set), try turning the dial, if it's not lighting up take out a 100ohm fixed, try again...if you end up overshooting between different fixed resistor swapping sessions and you'd like something more fine in between, start throwing resistors in parallel there.
A final thought, if you want, you can use a larger value pot afterall, and avoid the problem of not having much usable dial range by getting a multi turn pot - instead of just getting about 270 degrees of one turn like normal, you can get a pot with 10, even 20 turns for full swing, and that way you do get a lot of control once you find the region that works....so you can try a 5k or 10k multi turn pot maybe...
That's about all I can think of