Hi Carla and welcome to the board!
Yes this can be done and is a common practice. You need to provide an OnSubmit or OnSave handler and set the name of the file in code. There should be an example of how to do this in chapter 10 of my book. If you don't have that, it's pretty straightforward. Check out examples in the SDK, or ping us back, and we'll dig up some code.
Patrick Halstead [InfoPath MVP]
InfoPathDev
Patrick Halstead
Project Manager at Qdabra