No problem. You'll often need to rename your hosted files before uploading them to shared servers.
For GoZync 5.06 and earlier: the only file you CAN'T rename is GoZyncMobile, but since that file isn't hosted, that's OK. (If you're working with multiple mobile solutions, read this.)
If you rename GoZycHosted, be sure to update its External Data Sources path in GoZyncMobile to reflect the new file name. That means you'll want to upload a new copy of GoZyncMobile into GoZyncHosted for distribution. See "Preparing a new file for your mobile user" here" Distributing Mobile Files.
As of GoZync 5.08, renaming GoZyncMobile is possible. If you’re not yet running build 5.08, follow the instructions under VersionHistory to get updated.
First, rename your GoZyncMobile file.
Then, in your mobile file, change the value in the _GoZyncMobile__ custom function to the new GoZyncMobile name.
Also in your mobile file, change the external data source reference to match the new GoZyncMobile name. Do not change the external data source name or delete the external data source; instead, modify the file path for the "GoZyncMobile" data source.
In GoZyncHosted, click the “GoZyncMobile File Name” button in the footer of the configuration tab, and enter the new filename (without the extension). Click the button again to close the dialog. This name is a global field so make sure you change it while the file is on your desktop, not on the server, or the change won't stick.
Now, run the “Zync - Prep and Upload…” script in your mobile file to upload new versions of your mobile files to GoZyncHosted. On the Downloads page in GoZyncHosted, the newly named GoZyncMobile file will appear as a separate file from the original “GoZyncMobile”; delete the original “GoZyncMobile” here.