How can I get some research going before my Ph.D. program?

You should not be too worried about getting your name connected to working papers before a Ph.D. program. If the opportunity presents itself, then take it, but don't lose any sleep over not having a working paper (or more) when you enter a Ph.D. program.

If you really want to get started on a research paper, find a professor who does research in the area in which you are interested. See if you can help the professor with a research project. To help, means you do whatever he/she asks extremely well, without complaining, and without expecting to be a coauthor. If you do a good job on one, or a series of projects, the professor may be inclined to coauthor with you on another project. You should not expect a professor to be willing to put you as a coauthor on a paper just because you helped out. Being a coauthor requires a significant contribution of time, thought, and ability to a paper. Also remember, that before you have started a Ph.D. you have limited skills and ability to conduct rigorous academic research. Therefore, your main asset is to put a lot of time into the research process. Your one hour of work is not equivalent to one hour of a professors time and therefore you should not expect to be putting in a 1 to 1 ratio of time with the professor if you are going to be a coauthor.

Before you start your Ph.D. program you might consider trying to coauthor a simple piece of research. You may not want to target a top-tier journal with your first piece of research, but instead try to replicate an important study and shed a small bit of new insight into what the paper did or some small way the paper changed. Being exposed to the research process at any level (even if it is not a top tier article level) will benefit you significantly in your Ph.D. program.