I tried renaming a part its model name changed but the common name remains the same.

Assuming you are not using Windchill. When you rename in session and change the common name, you need to manually save the file. When rename in session is used it is not written to disk. are you certain you saved the file after changing the common name in session?

Yes, this is normal behavior. When you rename the file, it retains the original common name.

What I usually see is that the common name is "grayed out", meaning I can't change it directly.

If you insist on changing the common name, you can do so via a relation. The following will set the common name to be the same as the model name:

ptc_common_name = rel_model_name()

If you want to ensure that the common name always gets set to the model name, you could leave that relation in there, otherwise, once it has done the job you could delete it or comment it out.

Be aware that this might not be acceptable if you are ever using Windchill as well as Creo. I believe the common name is used by Windchill for some sort of function - I haven't ever worked with Windchill, so I'm not 100% sure of that.