It's not entirely out of the realm of possibility to do some kind of business during residency. The main question is what kind. The time constraints of residency are significant, even in an "easy" specialty. No matter how many hours you work a week, you DO NOT have control of those hours.
Note that working for Kaplan and starting a business are not the same. You could theoretically work for Kaplan... but- what will you do when you are on call the same day you are scheduled to teach a class? If you aren't done with your clinical duties before class starts? You can't leave your fellow residents to finish up while you run off to teach. Being late for class won't go over well, either. Q3 or Q4 call makes it almost impossible to be somewhere every Tuesday at 6:30 pm for 12 weeks.
If you are going to make money by trading your time for money (ie working for an hourly rate) you are probably better off looking into moonlighting options.
Some business are amenable to working around hours of residency. Like the blogging that Dr Kim described. From what I understand (I'm looking into this idea myself) though, blogging takes several years to develop any real revenue from, and that is with a lot of effort.
Other businesses, especially web based ones, could be doable. For example, if you know now to design web sites you could do that on the side.
I know I will get flamed for this but MLM is a legitimate business model. Yes, a most people who join don't make much money because they get discouraged and quit. You have to have thick skin to succeed and become very tolerant of hearing "No" as well as having people put you down while you are working on it. It's simple but not easy to keep plugging away at. There are tons of opportunities. If you can find one that has a product you like and would use anyway, that is an option, as it is totally flexible. Plus you get to be around positive people, which is a nice contrast to the constant negativity in medicine.
Whatever you do, realize that your residency is your first priority. Time in residency isn't just physical time spent in clinic/hospital. You also have to spend time outside of those hours reading/studying, preparing presentations, etc. Most specialties have an annual in service exam. And you will need to read up on the conditions your patients have. Plus you'll have to do some presentations: type and frequency vary with specialty and program. Plan on at least a couple hours a week outside your clinical duties for this.
