Just duplicate the comp in the Project Panel by selecting the composition and pressing Ctrl/Cmnd + D. This will give you a unique copy. You can re-name it and change everything in it without changing the original.
If there is a precomp in the original then changing something in the pre-comp will make a change in every comp that contains the pre-comp.
You can also duplicate the precomp, re-name it, and then replace it in the the timeline of whichever comp you want. You do this by selecting the precomp in the timeline, then selecting the duplicate pre-comp in the Project Panel, then while holding down the Alt/Option key, you drag the duplicate to the timeline and release. This is the standard technique for replacing footage in a timeline.
thanks very much Rick, that seems to work
so it looks like you do have to actually replace the comp (alt+drag), not just add a duplicate from the project panel--or else the animation/effects get cleared out?
IS there a way to do this via scripting? For example... I have a calendar grid formed by a solid square and a text object for the number. I'd like to generate these calendars without having to make a unique "day" comp for each instance.... does that make sense?