OK, I got some very interesting responses to my suggestion of an experiment that can be done for an intro physics lab. I think I didn't explain myself too clearly on the premise and how I'm going to present it on here, so I should do that now.
While I tried to be explicit in describing the experiment and what would be a good way to do it, I don't actually want to reveal the whole hand. That's why I went along with the idea that there could be a dependence of the period with varying weights, because that is a very likely path that the students might attempt. If someone is thinking of actually trying to introduce this experiment in an intro lab, I don't want the possibility that some student might google it and find my blog where the whole thing has been revealed. :) That would defeat the purpose of them doing this without any kind of "previous knowledge".
So while I'm trying to be as clear and complete as possible in the experimental description, and the "philosophy" behind it, I don't really want to reveal everything either. In fact, I'm hoping that there WILL be students who decided to figure out if they can do it by changing just the weights. I consider discovering something that cannot work to be very educational. In fact, in science, knowing what doesn't work can be quite important (re: Michaelson-Morley experiment). They at least now know that changing the weights would not work. If they are curious enough, they'll try to find out WHY it doesn't work, and this is where the physics can be introduced.
Note that the experiment that I had suggested does NOT require that they have learned anything in intro physics. It is quite independent of the lesson they might have received in class. So in principle, this experiment could be done even during the first week of class. It doesn't require that they had learned about simple pendulum.