I'd split this into basic and advanced:
Basic:
Mini phillips screwdriver set
Mini Straight screwdriver set
Good set of regular Phillips/Straight drivers
Set of both metric and SAE allen keys (ball ends are nice)
small machinists ruler
6" or 12" straitedge
light-duty metric/sae socket set in up to 3/4" or 20mm
compact multimeter
drill index in fine gradations from very tiny to at least 9/16"
I have a string height/setup gauge from Stewart McDonald that beats the pants off of a machinists ruler for checking string heights.
Advanced:
set of nut files in .96 to .16 (ballpark)
set of radius gauges
longer straightedge
shorty straitedge for finding high frets
set of "jewelers files" or hobby files
nut slot marking ruler
I like to have a few other goodies around, but these will get you through about any setup/diagnostic on a bass. If yo are going to be doing a lot of fretwork you will need good fret tools. things like radiused sanding blocks, fret files, polishing supplies... it can run into big bucks fast. But for getting the most out of the bass you want to be able to accurately set the action. I do almost all of this with a string height gauge, a radius gauge, and a set of allen keys. Depending on the bass you will need the proper tools to handle the truss adjustment.
Other stuff to have on had are consumables like cardboard, masking tape, and cloths to protect the bass while you are working on it, or super glue for instance. Things like a soldering rig are dependent on what work you are doing. I'd say that a good selection of wiring diagrams and a book on guitar wiring principles would be very helpful for troubleshooting as well as repair. There is a huge array of gear needed to cover every eventuality. Basic setup/repair is much different from shaping, routing, sanding, or finishing. Unless you are planning on making a lot of guitars I'd suggest getting as much of the finish work done prior to taking delivery of the parts as possible.
Pete