The problem with a lot of cheap cuts is that they are not really suitable for stir frying or other short cooking methods because they are less tender, more muscular, more sinewy. Instead, cheap cuts from e.g. beef often end up in pot roast dishes, where they can soften up and become really good.
That being said, a few good options for stir fry meats are, in my opinion:
- chicken thighs, i.e., dark chicken meat, which also usually stays juicier than breast. I can get them at my local supermarket, de-boned, in free-range quality, for quite a low price. Need to do a little cleanup usually, but nothing too bad.
- cutlets (schnitzel in German) from beef, veal or pork: tenderize them a bit with a meat mallet, then cut in strips across the grain
- ground beef: while it's not whole meat, you can cook ground beef with a little bit of oil on medium-high meat to get the water out, then brown it in it's own fat and a little oil, then finally add your veggies and other ingredients for a "fine grained" stir fry, so to speak
- other cuts of beef: Kenji López-Alt covers cheap stir fry cuts from beef beautifully here