Irvin Finkel is a truly entertaining and intelligent speaker. He has a bunch of videos and lectures on a range of subjects on YouTube through the British Museum Channel. I highly recommend not only his work, but the whole channel.
I was so inspired by that video when I first saw it that I built my own Ur board.
I prefer the 4 2-sided dice approach to the single 4-sided die, because the normal distribution of values allows for slightly more strategy.
We also house ruled that rolling a 0 gave you a reroll token.
You can make most oddly-sided dice as barrel forms (a.k.a. "long dice"). Take a cylinder of uniform density, and grind the ends down to cones, at an angle such that they cannot be a stable result. Now plane off the required number of stable surfaces at equal intervals around the barrel, and scribe its value on the opposite side. (Or for even-numbered sides, you can also construct it as an antiprism, capped by pyramids.)
If one cone has finger-grips added, the die can also be spun like a top. These are teetotums. A dreidel is essentially a teetotum d4.
For a d2, you add two parallel planes to represent the two stable results, and unstable-angled planes on the sides so that the die will always roll from them onto one of the stable planes on any flat, level surface. Or you can just curve the two sides, such that they meet at a point.
You can even make a (nontrivial) fair d1 with a clever arrangement of planes on the barrel. When rolled, the fair d1 will always come to rest on the same face, without being unbalanced, because all other planar faces are angled to roll the die to an adjacent face. I can't recall if dice of this type are dependent on being rolled in Earth gravity or not, but it seemed like they were dependent on angle of repose in some way.
Teetotums are great.
Games like TRGoU frequently use beans that have a spot on them.
In my case, I painted the faces of wooden craft cubes black or white, 3 sides each.
