For a while, I've had an idea for some sort of “measuring glove” that would use an accelerometer to measure distances; an invisible ruler, if you will. However, I couldn't think of a way to craft an interface that would be able to handle everything I wanted the glove to do in a graceful manner.

Then, Ashbad told me about KermM's Clove. The Clove is versatile enough to handle all the possible measurement types gracefully, and, from what I've read on its project thread, it shouldn't be difficult to remove several of its outputs, change the locations of some of the contacts, and connect it to an Arduino.

I plan for the Empiricist's Hand to be able to measure the following:
-Displacement between two points within a given frame of reference
-Change in velocity over an interval of time
-Average velocity over an interval of time within a given frame of reference
-Acceleration
-Change in acceleration over an interval of time
-Average acceleration over an interval of time
-Area of a quadrangle, triangle, or circle
-Volume of a quadrangular prism, triangular prism, or cylinder; possibly a cone or sphere if I can fit it into the interface
-Angle between two surfaces in degrees/radians
-Arc length between two surfaces at some radius

This is quite firmly in the planning stages at the moment. I'm researching parts and determining the hardware. The most likely candidate for the accelerometer is to rip one from a Wiimote nunchuck, although that option may lack the degree of accuracy that I'm looking for. Shelling out $35.00 for a better accelerometer is not the most favorable of options, but if the Wiimote's sensitivity isn't up to snuff, I'll probably have to.

Unfortunately, AP exam month is just starting to rear its ugly, headache-inducing head, which means that my progress on any of this will be minimal for a couple of days.

Ever since Motion Plus was introduced to the Wiimotes they've had a 6-Axis gyroscope. It's pretty likely a Wiimote will be sufficient. Now, the nun-chucks, I never knew had accelerometers. You could also probably find a Motion Plus Dongle (before it was integrated into the controllers) and take that apart. Not sure how much they run.

To calculate distance, I assume you'll use speed over time? Are you going to assume the speed won't change or are you going to sample speed and time constantly and make calculations based on change of speed?

You're describing an IMU. There are lots of options available, for example this fairly cost-effective unit from Sparkfun. A purpose-designed IMU will probably work much better and be easier to get running than ripping gyros from some other piece of hardware.