What makes them suitable - You'd want to focus on cuts that are especially lean, without a lot of connective tissue or fat, marbled or otherwise. Raw fat is probably unappealing, flavor-wise, and that's what often causes meat to go rancid, so focusing on cuts with just muscle tissue, and minimal marbling is what makes it good (and safe) for tartare, I'd think.
Cheaper cuts that are good for stews or slow-cooking often have a good bit of connective tissue in them, which would make them poor choices. Unless you like a gristly tatare. :D
Probably round or rump, and sirloin - all already mentioned by you - would be decent choices for those reason. I'm not sure you'd need to look further. Of those, sirloin has the reputation for having more "beefy" flavor, but round and rump are often among the cheapest cuts.
EDIT - UK and USA cuts by the same name are not, apparently, identical cuts. Please see the comments to this answer.