Clamp - $5
Clamps come in a huge range of size and styles and chances are will be reused across a number of different jobs. Small clamps can be used for smaller wiring and solder tasks. Whereas medium and larger clamps are useful for holding components in place when cutting or even painting things like PCI and 5.25" bay covers.
Paint Applicator Gun - $10
This ingenious tool makes using spray packs a little bit easier. This device fits onto spray cans and uses a trigger handle that makes even consistent painting graspable for even the worst painters. I have also found that these types of paint applicators prevent hand cramps from extended painting sessions.
One of the most common and price wise mods today is cable sleeving. Cable sleeving is one of those tasks that is easy to learn and impossible to master. A low cost way of instantly changing you cases interior looks and although being time consuming the end result can be the make or break factor on larger builds. Sleeving requires very few tools besides your own two hands and tools that do have to be purchased are fairly inexpensive.
Crimping Tools - $10
Whether you have a broken connection or are extending wires you will need a tool for crimping on the connectors. It's a simple to use tool, which can be substituted with some small needle nose pliers for modders on a budget.
Wire Strippers - $10
Used for removing the plastic casing on wires. Sure, you can fluke a perfect cut with your scissors or knife, but for the sake of a few dollars, why would you risk damaging wires? Available in a range of styles my personal favourite is the old school clamp style.
Heat Gun - $35
Heat guns are used for heating up heat shrink allowing it to shrink down and hold sleeving in place. They are fairly cheap and usually have settings for two temperatures. I have seen girlfriend's hair dryers and lighters used for heat shrink before, but you just can't beat the control of a heat gun.
The other benefit of a heat gun over other "tools" is you can use it for acrylic work like bending if you ever plan to travel down that mod path. Re-enacting scenes from movies with heat guns is another favourite of a few modders I know - only when they think nobody is watching, of course!
Soldering Iron - $15
Essential for joining or re-attaching mistreated wires, soldering irons are readily available from most hobby shops and deceivingly easy to use. Paired with some good quality solder wire and some old PCB's, soldering is a skill you can teach yourself from a guide in a few hours, and is invaluable for cable modding.
Multimeter - $10
As with all mods and customisation work there is always a risk of damaging your components. If you have previously learnt to solder, chances are you can fix the problem yourself. A multimeter is a handy tool for monitoring voltages and electrical values on components. Most multimeters are straight forward to use and give you a range of readings and outputs for whatever components you suspect are faulty from wiring to LEDs.