There have been a few of these posted here in this condition that we've been able to ID based on the shape of the frame and the left side plate. But as a general rule, they are pretty much impossible to get a 100% ID on. There were hundreds of different pistols like it made by dozens of companies big and small. If it had a name on it, it was likely on the barrel, and in that condition there's no bringing it back. Best I can tell you is it's most likely a .22 short, and probably 1870's or 80's.
I would not recommend electrolysis on it, that barrel looks like it would just crumble away. I think I would pick the loose rust off with a toothpick and soak it in cider vinegar for a while, and keep picking with the toothpick until it's reached a point you're happy with. Then wash it clean with alcohol or acetone, dry it in the oven for a while to get any residual water out of the metal and seal it by coating it with a thick layer of wax (I use toilet wax rings for this) and bake it again for a while to let the wax penetrate.