Little sparse on details on what to suggest; based on the fact that you have a duplicate naming issue it would be presumed that these two tables (all tables between the two databases) are exclusive of each other i.e. you are not appending rows between tables, but are appending tables into a singular database. Or?
Normally within the Access IDE, the renaming of Access level objects is fairly reliable (though not full-proof); the main issue will be dealing with code behind issues on the VBA side of things (i.e. hardcoded strings used everywhere vs. contant declarations).
You've probably have had to resolve this doing the rename approach (based on the staleness on this posting) but I suggest a rename that applies just a suffix or prefix to the original name.
BTW: did the coding/calls in either of the databases call into each other and to these two duplicately named tables? The name collisions must/might have already taken some approach for uniquely distinguishing between the two tables within the same database; something you might want to emulate in your renaming.
