My in-laws have an older home and the back patio has those rocks embedded in the concrete that a lot of older homes in Texas have. They are very annoying to walk on. My son and their dog just flat out refuse to walk out there. So 1) Does anyone know what this is called? and 2) What would replacing this with a good old-fashioned concrete slab cost? Is it more cost-effective/even possible to just cover the rocks with concrete?

If I am thinking of what I believe you are trying to explain its just exposed aggregate concrete.

If it is, just pour a thick layer of self leveler over the top of it. That's the cheap and easy way

Yes, that is what it is called! Thank you! I have been trying to Google this for 3 days and couldn't come up with anything that wasn't landscaping rocks.

Travertine ut

Also can be called chattahoochee. I hate it as well and we have couple thousand square feet of it.

Smaller area self leveling seems like a decent solution. Or you could top it with an appropriate tile.

You can do what we call a "concrete overlay". As long as you match the joints of the old surface in the new, you should be fine. I'd expect the new concrete to bond to the bottom pretty easily as long as it's been power washed and cleaned. Having the old surface damp, but no standing water is good. During the pour and while it's curing, protect from wind. As it hardens, I would probably use a water cure by soaking the concrete in water for a few days. The biggest risk is the surface curling from bad curing as it sets.