Hi Kerry! I know I always say this but only because it's 100 percent true....no one method of weight training will work for everybody. You have to find what works for you and then continue to make changes (sometimes subtle and at times even all out radical) to keep the results coming.
One body part per day can be used in so many ways. For example, as Lynne said below, its great because it allows you to focus and give your all to that one body part since you know that your workout is going to be short and to the point. The key is you HAVE TO give it your all. And as mentioned, the recovery time for the muscles with this approach is nice and long to welcome the healing and repair for that intense workout. You may also want to do one body part per day at less intensity as a beginner who is looking for a short workout that is not overwhelming. Or perhaps an exerciser doesn't have a lot of time to workout, or is on vacation and looking for a quick fix workout to do without giving up too much vacation time.
As with any training program, you'll see enough results by 4 weeks to know if this is something you'll want to continue with, alter slightly (ie: put one total body workout in the mix to just hit everything again lightly that week), or do away with as a regular rotation.
Keep in mind no matter what your preferred routine, change is good. So even if you find a training mode that you simply love, you will still need to change it up somehow to keep your body from adapting and getting too comfortable with the current program. Subtle changes are mostly all you need to keep the body guessing, but every now and then it's "shockingly" valuable to just do something totally different for 2 to 3 weeks.
From my personal experience, I have found the one body part per day to work best for me when I am at the top of my game, seasoned and ready to meet higher challenges. I put a new twist on my body part exercises and really go for it. After about 6 weeks of one body part per day I'll go to a three day split (or 4 day split) for a while. Then I'll go back to one body part but with new exercises, but this time every other week I'll throw a total body workout as a bonus workout. Then I may do total body endurance workouts for about three weeks to let the joints and ligaments and connective tissue recover before going back to high intensity.
I hope this gave you a little insight as to how you have to find out what you like and how to constantly manipulate it so that it stays fresh and challenging.
Good Luck