When you rename an item, if it is a weapon, when you kill something with it the custom name will appear in the Kill Feed. Also if you click on the item in your inventory, you will see the custom name instead of the normal name.
If you die, and drop the weapon, if another player picks it up, it will still retain the custom name until they rename it or someone else does.
Rename an item by selecting it in your belt. If it is an armor piece, or other item that is not normally selectable, you must first select a weapon, then move the new item into that slot. This should now have the item selected.
If You wanted to name your item: "Cool Item" then do the following:
Once the item is selected, type: /rename "Cool Item"
Notice the new name is in double quotes. This is important.
You may also add color to the name. So if you wanted the name to be yellow, you'd instead do it like this:
/rename "<color=yellow>Cool Name</color>"
The </color> tag ad the end is just as important as the double quotes.