1 - 2 mm is good. I use soft leather, not the hard kind.
Trace a record, including the hole, and cut out.
Put the mat on a 'table, and use a felt pen to trace a perfectly centered outer edge as it spins.
I've tried both sides, and have not been struck by much difference. But once you try it, you'll never go back to felt.
Alec, a Linnie mate of mine made up some leather mats - which are streets ahead, soundwise, of the previous mat he'd experimented with (some sort of dense foam which he can no longer get). Which back then was ahead of the Linn felt mat.
He played around with several types and the best one turned out to be a tad over 1.5mm thick, with the smooth side (outer skin) against the platter and the "suede" side up. But he used up quite a few skins before he found the optimum!!
) (He's obsessive!
I'd post a pic but I have no idea how to!