Thanks to our Ausie mates for playing, but those aren’t the right things.
The items (I believe) the OP refers to are neither for mourning, nor sweatbands. Both those things are common in US sports, but that’s not what (I think) the OP is talking about.
I too have always wondered what they are. Here’s a more detailed description …
They look as if someone has taken ordinary shiny plastic electrical tape about 1" wide & wrapped it around their bicep, either at the largest bulge, or more commonly, just above the elbow. It often looks like the tape is quite tight with the muscle bulging out on either side of it. We can’t tell from the TV how many layers there may be, but it’s always just one wrap wide, about 1".
Some players wear a an elbow protector made out of neoprene (wet suit material). This is a (usually black) tube which extends a few inches above & below the albow. The OP’s armbands are not that either.
I have also seen charity advertising bands. These are not those either. Only a few players on any team wear the tape-like bands I’m talking about. And when they do wear it, it’s on both arms.
So, any fans of US football know what these are?