Okay, how to rename your pet? Savefile editing. That is the only way I know of. It's not that hard, but some may find it daunting. How to do so? I'll tell you...
1 - Most importantly, before going any further you should make sure to backup your savefile before you begin editing, just to make sure. (Make a copy in another folder.)
2 - Open your savefile (not the copy you made) in a text editor, and even Notepad will work for this.
3 - Do a search for your pet's name. In your case, Harvey. Since your pet's name matches an NPC, you'll want to make sure you have the right one. It should look like this: <NPC xsi:type="Cat"><name>Harvey</name>
4 - Change where it says Harvey to what you want your pet named.
5 - Save and exit.
6 - Open your save in Stardew and verify it worked.