Does anyone know what this guitar accessory is called....?

They're called a locking nut.

It's used a lot in guitars that have floating bridges, like the Floyd Rose, because guitars that have floating bridges go out of tune more often, those locking nuts are necessary.

Unfortunately you cannot just buy locking nuts and just place them on any guitar. Take for example a Les Paul, which just uses the standard nut and has no holes to put the locking nuts in.

Or it might be a capo :/

A capo is used to clamp all the strings across one fret so the open notes become whatever that fret is, but it's not used to keep the guitar from going out of tune. However, there is a locking nut that is designed for keeping the guitar from going out of tune. Both can be purchased at nearly any guitar store but the locking nut would have to be matched to your guitar.

I believe what you are looking for is a capo. Doesn't keep the guitar from going out of tune, though. You should be able to find one at a good music store where they sell instruments.

It's either a capo or a string lock. Nothing else makes sense.

sounds like a capo. it's not to keep it in tune, but to transpose the key. you can get them at music stores where they sell instruments and strings and stuff.

if your looking for a cheap alternative to stay in tune, wrap electrical tape the headstock right after the nut. works amazing on a bass as well.