I find that for most things a simple, single small electrical screwdriver will do
However my main/normal kit is one of the "computer tool kits" in a fake leather type case from memory
1x large screwdriver handle with reversible torx bit (some companies still seem to like the security given by using torx headed screws/bolts, and at one time they were fairly common with several brands)
1x large screwdriver with reversible screwdriver bit (phillips on one side, flat on the other)
1x Small phillips screwdriver
1x small flat screw driver
1x insulated tweezers
1x chip puller
1x screw grabber (useful for getting errant screws out from the side of a motherboard etc, or for starting motherboard screws)
1x tube with assorted screws/motherboard mounting bits.
I added a small tube of artic silver, dropper bottle of iso propyle alcohol and several blank cards (the sort MNBC used to send me every couple of weeks to try and get me to take out a card with them

) to that set for heat sink cleaning etc.
In addition to that I've got an anti-static mat and wristband I use when building machines as opposed to just swapping out a part (I find it easier to lay parts out on the mat and know i'm always grounded when building a full machine).
That all goes into a small case along with a basic network cable tester, short network cable, a DVI-VGA adaptor, and a couple of USB-PS2 adaptors, crimp & punch tools, and a few RJ45 connectors.
Most of the time I'll just use the small screwdriver I keep on my desk, but depending on what needs to be done I'll either grab the small soft case, or the full case

if I'm going out of the house to do anything to someone's computer the full case goes - I got fed up of finding that what was described as a simple problem ended up requiring me to head home to pick up a different screwdriver or network cable etc.
About the only other tool that sees regular use when I'm working on computers if the mini maglite on my keyring, usually to find a screw I've just dropped under a chair...
That doesn't cover the boxes of things like spare NIC's, old CPU's, fans, heatsinks, cables etc that seem to take up far too much room but are handy at a pinch.