Looks a lot like the spotty-leaved variety of Aucuba japonica. I've seen it outdoors, but that was in a reasonably warm and frost-free climate.

Codiaeum variegatum 'Aucubaefolia'

The terminal on the left shows opposite leaf arrangement, so I'm going with Aucuba.

Sounds good. Isn't it amazing how few photo's clearly show leaf arrangement?

Yes, leaf arrangement can be deceiving if leaves have fallen off in some spots or if there isn't a good side view.
You can root Aucuba in water. Before moving to NC, I took cuttings of a few different cultivars and after they rooted in a vase, I potted them. I've seen Aucuba as both a house plant and a garden plant in PA, zone 6. The gardener from whom I received my original variegated( female) plant had a large potted plant inside her home. There was also the parent plant outdoors, against the house in the shade.

Stem structure and leaf venation is not that of an aucuba. My first thought was croton, but I was not sure.

Thanks, all! Now that I know it's most likely some type of aucuba I can figure out what I'm going to do with it care-wise. Apparently they're easy to propagate via cuttings, so I'm probably going to have fun trying that out at some point.

Hurrah for the plant nerds :)

Looks like Gold Dust dracaena, Dracaena godseffiana.

It looks like alternate leaf arrangement to me, and the habit looks more like a croton.

I've had a good look at some Aucubas around here and the stems are smooth and green, even on quite old parts of the bushes. So, on balance, I lean towards the Croton too.

Whatever it is it's not very happy. I think I'd take some cuttings and start again.