The only option would be to make an exact copy of that unit in the object editor with a different name on it. And then trigger it to make it look like he changes name, by replacing the old unit and then select the new one for the player. If it is a hero you need to transfer the items thought.
The easiest way, I think, is using the Berserker Upgrade. I have never done this before, though, so I can guarantee that it works.
Make one costum unit with the first name, and another unit with the same stats, but with the other name. Then base an upgrade on "Berserker Upgrade", and change "Data - Effect 1 - Unit Type Affected" to the first unit, and "Data - Effect 2 - Unit Type Affected" to the other unit.
And when you want to unit to change name, just give the upgrade to the player owning the unit.
I'm not into upgrades, but if you unlike abilities can have two upgrades based on the same base upgrade, you can make an inverted upgrade to change back. I hope you're not planning to change too many times, since you will need one upgrade for every time you change...
Ghan has said he has fixed this. Monovertex please confirm this fix. This was only a problem with people that had signatures in the upper levels like not the special members but the respected members.