Is it possible to rename a core module so that I can use it multiple times on a site?
The reason I want to do this is then I could change the resource file for each instance of the module and it would not effect all instances.

If this is possible what do I rename?

I'd be careful with that. Depending, each "version" would also require its own database tables. That might be a lot of work for a little benefit.

Which core module are you referring to?

I want to have different instances of the file Repository module on my site but when I change labels in the Language | Resources section it changes the labels for all the different instances of the Repository module I have on the site.

If you have the time look here  www.salmovillage.ca and check the Government tab on the menu.

Under that tab I have Council Meeting Minutes and Bylaws and a couple others.   The office staff upload PDF's here for public viewing.  I would like to customize the labels that head up the columns.   For example I want to change the word FILE to MEETING DATE on the Council Meeting Minutes page but I do not want that label on the other instances of the repository module where the Bylaws are listed.   

These labels can be changed in the resources file but like I mentioned the changes made there affect all the instances of the module.  

I thought if I could simply rename the module then it would be recognized as two separate modules and do what I want.  

This could be worth asking the Repository core team in the dnn forums, they may have  a quick tip that would work, or like Joe says it could be a lot of extra work. . . .

Thanks for the answers,

I asked over at the DNN forums and was told that the repository module support multiple templates and all I would need to do is duplicate the template? Then I could change the resource file for each template.
I did this and was able to successfully apply the new template but it appears that the column headers are contolled by the repository resource file and not the template resource file. Back to square one.

ok, thanks for the feedback, I would make sure you inform them of this and see if they can provide a better solution - feedback like this helps them to keep improving the module,