Beyond the usual pliers, screwdrivers, electrical tape, voltage meter, and spare lamp parts. There are a few special tools in my tool chest: various sized clamps and wire stripper. The locking clamps are handy and inexpensive. When you need to get in a tight space or have hold of a cord without the extra hand. These clamps can hold on to that part while you need to make an adjustment. Since lamp parts are a little more delicate than other mechanical parts, I find myself using these clamps more often than the rough pliers. These are pretty common in hardware stores and may be sold as set. The wire strippers are a nice addition to a tool box. This set strips and pulls away the excess cord for easier removal.
lamp repair are probably only found in specialty tool shops or lighting shops. In this photo from left to right I have a pick for getting glue, paint or gunk off a lamp. A socket cap wrench that locks into a socket cap for tightening. Pinning pliers for prism wire and specialty jobs. Chain pliers for spreading chain or closing chain; these are better than channel locks or regular pliers since they won't scuff up a chain. Across the top is a dial caliper for measuring inside and outside diameters. This helps when I am looking for something in the spare parts section. I can measure the lamp and only carry the caliper to the spare parts or search online for parts and know the size I am looking for.