Welp, if it's not a hassle for DMs to respond to the requests in a timely manner, then that removes the impetus.
But I don't really think it would require sorting through players' inventories and looking a descriptions because...who cares what I have called something in my inventory, if only I see it?
The only time it matters (really) is when I show the item to someone else. And in that case if I'm cheesing, it could be reported...just as if my PC does anything else in RP that requires reporting or a ruling.
If you don't show me your belt you have named "Girdle of Storm Giant Strength," why do I care what you (personally) call it? It has no effect on anything.
The ability to name items what we like is largely personal, like an in-game journal entry IMO, with only actual player interactions being something that DMs should have to spend time on. (In two years I've only asked for this once, so it's not that important to me, but I hate to see DMs' having to spend time on things like renaming food items...and players having to post such requests and wait around in game for them to happen.)
I'm not being dismissive of such a request, but rather saying that I see nothing wrong with allowing the player to call the item in her inventory the other name without a DM having to be involved, just as, for example, we might RP adding honey and cream to a cup of tea even when we don't have either in our inventory.