What you want is real computer control, with closed-loop feedback for precision. Good news: it's super easy to connect an Arduino board to the OWI arm and let it do all the work! You won't even have to take the OWI arm apart or damage it.
Here's a video showing it in action, under fully autonomous Arduino control:
Here's what you'll need.
1) OWI Robot Edge Arm ($37)
2) An Arduino Uno board ($30)
3) An Adafruit Motor Shield ($19.50)
4) One potentiometer and at least three little trimpots ($5)
5) Some perfboard to mount the pots ($5)
6) Some jumper wires ($6) and pin headers ($7.50)
The total should be around $100. But you may have some of this already, and if not, you'll be able to reuse these boards and other parts for countless other projects.
Step 1: Connecting the motors
Just use a little solid (not stranded) wire to attach each OWI motor connector to a Motor Shield output. Strip a quarter inch at each end and plug one end into the OWI Robot Arm motor connectors and screw down the other ends in the Adafruit Motor Shield connectors, as shown here.