There are a few possibilities but its fairly easy to isolate the likely problem.
Unclip the noodle (where the cable comes out) and check if the problem is in the V brakes or the cable.
If its the V brakes, it could be a broken spring, or the brakes need oiling.
If the brakes are free to move and spring back firmly, its likely a problem with the cable or the brake lever. Without pulling the cable out of the housings, slide the cable back and forward - it should be free to move. If not, check the brake lever is free, or it could be a corroded or kinked cable. If its the cable, a replace is only a few dollars and is the best option (I almost always put in new outers as well). You can try adding some lube to the cable and working it, its usually successful but never as good as new cables. If the cable is kinked, you may have luck straightening it, but again, a new cable is always a better option.