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How do you determine the amount of Stress Dice when making a Supply Roll for Time Countdown?

Would it be based upon the character with the highest Stress Level?

Depends upon the amount of time... and whether you're dropping one per facehugger or one for any number of facehuggers.
The latter is easier to calculate the expected time
1d expects 3-4 rounds (Technically, by the math, about 6 rounds, but in practice, it's about half that)
2d expects about 2 rounds more
3d - 4d also about 2 rounds more each
5d and up about a round each

But it can vary WIDELY. what it does do is set a hard minimum number of rolls, so it's less tense.

If using the drop a die per facehugger, it's about the same for 1D: due to sequencial issue, it's about 3-4 rolls expected... but each additional die only adds about a single roll to the expectation. The actual math is ugly as hell, I did it once several years ago... I lost the spreadsheet.

Hi everyone, I’m new here and trying to get my head around the Time Countdown mechanic. When setting up a countdown for a custom scenario, is there a recommended way to choose the initial number of Stress Dice based on how long you want the countdown to last?

For example, if I wanted the countdown to create tension for roughly 5–6 rounds, would starting with 2–3 dice be reasonable, or is it better to test the expected number of rolls through playtesting?

If you want 5-6 rounds, 2D, dropping one die per facehugger, should be about right... but anything from 2 rouds to about 18 is within "statistically not unlikely" - 1 in 36 times it will be 1 round. 10/36 will be reduce by a die.
1d is of course 1/6 chance.

You can't "dial it in" to 5-6 rounds tightly using the 1 in 6 chance.

The odds of X rolls to go off on 1d:
1 roll 1/6
2 rolls 5/36. (the 5/6 of not on roll 1 and the 1/6 of round 2)
3 rolls 25/216 (the 5²/6² of two fails, and the 1/6 of 1 hit)
4 rolls 125/1296 (and so on)
5 rolls 625/7776
6 rolls 3125/46656
So odds of 1d making it to 5 rolls or 6... due to the sequencing issue and the long tail issue, it continues to get thinner and thinner.
The odds of resolving at either 5 or 6 but not earlier are about 6875/46656 or about 15%... and the odds of it going off in rounds 1-4 are about 671/1296, just about 52% so, half the time, it's under 5. and about 35% it's over 6 rolls...

2d gets far uglier, because of the branching in permutations but given the 10/36 chance of dropping to 1d and the 1/36 of ending it's about 2 extra rounds... I'm not redoing the math overall, but 1/36 times, it's one roll to done, 10/36 it's reduce to 1d, and 25/36 it's 2d next roll

If instead simply reducing the timing pool by 1d if any trigger (but only one less die), 2d gets easier
1 roll 11/36
2 rolls 275/1296
3 rolls 6875/46696
Then into 1d odds for the remainder

You can't get a simple answer for "dice for 5-6 rounds"... but the odds of "1-6 rounds on 1d is about 67% of the time"

I'd personally set 1d and hope... as 5 rolls or more is less than half the time.