I believe the ability to do this in the EC was broken sometime after the launch of SA and it was a client feature not a mod/skin. I recall reading a post about it (perhaps the tamers forum?) not currently working in the EC with no ETA.
Happily, this is not true! I went and checked today just to be sure, and although it's a little subtle, the renaming is still there... at least in the version of Lucis I was using (Designed for 4.0.7.4.)
Under "User Settings" > "Options" tab, switch to the custom UI for Lucis. Open your Pet Gump as follows;
Now click the Yellow globe. This trick only seems to work from that option. Give it what ever name you like; for the fire beetle it used to be "Appletize (My character) Goes Down In Flames" because for that character I like that message to flash up when he get's killed and dismounted... His other pets are "Appletize Dies Again" and variations of

But anyway, just to prove it, I took the "s" off the end and renamed to check it works, and yes it sticks.
Just be aware that occasionally this custom UI adds a space at the start of the name, so delete the lot and write from blank to hit the maximum character length (which is maybe 1 character more than the above...)