You dont have to change the product name, although if you do it shouldn't matter either. Apps are identified by their bundle identifiers.
Likely what you really want to do is change the display name, which is the 11 characters that appear below your icon on the device. End users generally dont see the "product name" as that is used for the executable or '.app' file name. In which case users will only see it if they look for the resulting .ipa file on their harddrives after purchase form the store.
In your info.plist there is a field for "bundle display name". This is what determines the name displayed under your icon. You can just change it there in the info.plist, keeping in mind that anything over 11 characters gets truncated. You can change this without an issue from apple, although your users might be slightly confused if the name is drastically different.