When I weave rigid heddle and tablet woven bands for the sale box I take to events with me (in non-plague years!), I mostly use Drops Karisma DK. It's 100% wool, 100m/50g and comes in a good range of colours. It's also pretty economical, especially if you buy it during one of the Drops discount events that happen several times a year.
I like to use King Cole Merino Blend 4-ply when I'm weaving trim. It's 100% superwash wool (meaning that it's machine washable)and 180m/50g. It's the yarn I used when I wove the samples for Noughts and Crosses and Tic Tac Toe and makes a lovely trim, as it makes bands that weave up fairly fast, but are less bulky when stitched to a garment than DK.
For the samples I make for my books, I like World of Wool Weaving Yarns, although at the time of writing, it's out of stock on their website. I really hope that they're able to get more of it, as it stands up to tablet weaving really well. It's 100% wool and 8000m/1000g and comes in black, white and orange. To extend the range of colours, I've been using the white yarn for natural dyeing. For one of the samples for my book Tablet Weaving in Theory and Practice: Double-Face Inscriptions, I used Brazilwood dye and an alum mordant from George Weil to get the lovely red colour that I needed.
Lastly, Piper's Silks. For very special bands, I use their 80/3 spun silk, which is 75m per spool. It's beautifully smooth and because of its tight twist, it stands up to tablet weaving really well. It's pretty fine, so when I use it to make a belt, I laminate together several layers of buckram, then sandwich that between the band and cotton twill tape and stitch them together. This yarn comes in a great range of vibrant colours and the woman who runs the company is an absolute pleasure to work with.
I've made a new video and added it to the beginning of the tutorial page here. It's all about the things I consider when choosing yarn for a new band.