It is low voltage wire de no big thing. Hard part is finding the spot you want it then getting the wire to it> That wire is easy to wire nut together ...........Is the thermostat on a second floor and if so are the floor joist exposed below?? If so it would be easy. Just drill a small hole with a long bit like 2 footer through the floor right below where you want to place it new, Then go down below, find the bit and drill back up through the floor going over like 2-3" assuming you drilled the bit right next to the upper wall. You can go through the bottom plate of the upper wall. You're gonna have to cut a thermostat box hole then use a small fish tape they call them go up through the hole below you drilled and find the end next to the hole ya cut in thew sheetrock. It can be done. You will need your wife to spot the fish tape when it gets top the box hole in the sheetrock.........
And there are a lot of vids on you tube about this . a picture or vid is worth a looo words. Hope this helps..........
Another way is cut the bottom 6" of sheetrock off the walls where you want to move it. Do all your drilling running wire and such through the studs down there. Then put back on the sheetrock strips you saved , tape mud sand and paint. I would feel confident you could do it this way. Again vids on tube LOL good luck buddy......

Realist - Everybody in America is soft, and hates conflict. The cure for this, both in politics and social life, is the same -- hardihood. Give them raw truth.