Well, it all started back in the days (circa 1991) and went something like this:

There once was these two MODEMers and they went to high school (how odd). Well, they met each other and became good friends because they had so much in common. They both liked to call all sorts of BBSs all over their NPA, but [person 2] didn't have a good handle to use on any of the systems. Now [person 1] had many handles that were variations on the same basic idea, such as Dr. Data, Dr. X, etc... Well, as you can probably guess, the Dr. part of Dr. Cheap comes from this chain.

So you are probably asking yourself, where did he get Cheap from?
Is he a cheap person?
Or is he just a person who wants to become a low cost doctor!?
Answer: D) None of the above.
So where did it all arise?

This brings us back to the two MODEMers, [person 1] and [person 2]. Well, many times when the two guys were just hangin' out, they would make comments on stuff like any young high schooler does. At the time [person 2] had a favorite saying about things that were either stupid or unfair...
"HOW CHEAP!" And thus you can assume is where the Cheap comes from.

Well, [person 1] noticed this was a favorite saying, and knew that [person 2] needed a good, original handle. So, he combined the first half of his own , and the second half of this saying, and came up with Dr. Cheap.
[Person 2] loved this handle... It was very unique, and had a sort of "ring" to it. So he stuck with it, and has used it all over ever since. It has been a conversation piece many times over, and everyone wanted to know the story. Enough people, in fact, that he got tired of telling the story, so he decided to "publish" it and release it on his WWW page!