I have tried several names and none have worked. It says “that name contains invalid characters. please enter a new name”. I have tried Loki, Persephone and others like it but all fail to work.
The error message may be misleading. I think the issue is that certain prominent mythical names are in a reserved word list so unavailable for pet or character names. (I know from experience that this is also true of most titles of royalty or nobility.)
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This, naming after known characters is not allowed, something like Loki for example for copyright reasons, there are however some pets like Skoll which can be tamed with names similar to or identical to known mythological animals/people. This is the only way one would be able to name their pets something that goes against the naming policy is if Blizz themselves has it on the nameplate.
It was a location issue. I hearthstoned to Dalaran and it worked fine at that point. Thanks 
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Well, that sounds like a legit bug, since I know of no reason pet renaming should be location sensitive.
Sadly this bug still persists. I could not rename my pet in Valdrakken(or at least near the stable master there) so I hearthed to Org and it worked no problem. Bizarre.
My hunter has a pet he tamed a while back in the Broken Isles. Busy with quests I didn’t rename it at the time. I hadn’t summoned it for a while so I’d forgotten it hadn’t got a name. Last night I was sorting my stable and decided to name it. The option for renaming didn’t show up in the menu of things you can do with it, so now the poor creature is stuck with the generic name it got given on taming. I’m not happy, nor is my hunter. His pets are his family.