Somehow, renaming a public function name messed up local variable names. After every rename, I make sure that Rubberduck is refreshed. Still this problem occurred, resulting in a difficult mess trying to figure out what the original code looked like ;-(.
It pasted a new function name right into an existing local variable name and did not change the function name in that same procedure as the local variable name.
Renaming feature suffers some serious issues:

I tried to rename "strFileName", but somehow Rubberduck thinks I am in the shtPatData and I want to rename that sheet name. Unfortunately, the renaming feature is barely useful this way.
Is this 1.4.3 or 2.0rc? We fixed thousands of bugs from 1.4.3 in 2.0, so this may not be an issue any more.
Looks like 2.x given the Rubberduck statusbar.
Ah, you're right.
Sorry, this is the 2.0.9 RC version. I use it with MS Office 2013.
Pretty sure the issue on this is in RubberduckParserState.FindSelectedDeclaration - overlapping declarations are using a naive approach based on the location of the selection. More correct behavior would be to find all of the overlapping declarations and select the one with the smallest scope.
Pretty sure this is fixed by now. Refactor/Rename was essentially rewritten since 2.0.9, and performance dramatically improved since then as well.
Yup, consistently using the ModuleRewriters definitely fixed this.
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Pretty sure the issue on this is in
RubberduckParserState.FindSelectedDeclaration- overlapping declarations are using a naive approach based on the location of the selection. More correct behavior would be to find all of the overlapping declarations and select the one with the smallest scope.