My problem with this is the name would only show up when you've already opened the chest. And at that point, you already know what belongs in the chest, because you can look at the items themselves.
Having it hover above the chest would be kind of cool (like a smaller version of the "you caught a fish" pop-up), but then everyone with a vertical set of chests (which if you're on Console then I highly doubt) would groan.
And at that point, you could use chests as text signs, although you'd have to cover them with something if you didn't want to see the chest.
Personally, I don't think I'd find this feature particularly useful and/or necessary, especially given that time pauses when you open chests (in Singleplayer) so opening the wrong chest doesn't really set you back, given that your chests are close to each other.
I think more players would be excited if you could lock signs or other furniture, so that you can use them for organisation without having to be super careful around them.