Hello there!

I have a small tricolor sedum that I split into cuttings. One of the pieces I broke off (I didn't even realize it the day I split them) is completely solid green, and continuing to stay, and grow, that colour. The rest of the rooted pieces are exactly like the parent plant still.

Did I accidentally make a new variety? Or re-make an old variety? :P

It sounds like you have a "reversion" so most likely it returned to the original non-variegated plant.
(I guess there could be, theoretically, a small chance of it having formed an all-green new mutation but you can't really know without DNA testing and seems less likely.)
The solid green plant will be more vigorous than the variegated ones. That's why it's good to keep all-green reversions cut out of a variegated plant so that it doesn't overwhelm the variegated parts. I'd give the green cutting its own pot so you don't have that problem.

to ATP!

Yes, it has it's own pot. I'll keep an eye on it.

Thanks for the welcome!

I just got some of this in 2 different types of sedum! kamtschaticum and sieboldii I'll take photos next week, they're both doing so great! I got them last fall and babied them through our Seattle winters (which aren't usually that bad - wet and lowest temps in the high 20's) but this winter lowest temps were in the teens for almost 2 weeks, and yet these both did great, losing some foliage but not all, and as soon as temps warmed back up, they began growing!!

Hi Rashputin, to ATP.

Tri-color is noted for have stems that revert to green. Danita's advice is wonderful. Keep any solid green shoots pruned out.
Tri-color will also have more variegation with different growing conditions. You can see here the wide variance in colors. Sedum (Phedimus spurius 'Tricolor')