Are you sure you need to copy the Sharepoint site or is it simply that a List or Document Library has reached it's maximum limit? If you really do need to copy the site, check out 'Save Site as Template'.
For instance if it is a Document Library, maybe the number of documents is at the limit.
Otherwise if it is a list, they also have a limit of the number of items that a single list may contain - something like this picture of the List Settings screen for example:
If so, you could simply create a new Document Library or List. For instance if your Project Document Library has 5000 documents (and this is the defined limit) and it contains Project 1 - 1350
you could rename it as Projects Document 1-1350. Then make a new Document Library called Project Document 1351-x