If you run the Standardize Layer command on any data, it will keep the name of an object if it has one already, but if it has no name then it will take the name of the Layer that it is placed on in the standardization process (if you check the check box that states Uniquely name unnamed lines)
Note that in the Standardize Process, the default behavior is that all lines on a layer of one object that is selected will be selected together and relayered to the Target Layer, however if you use the CTRL key while selecting you can select individual objects or a selection of individual objects and send them to a Target Layer - allowing you to separate data that happen to be on the same layer during the Standardization process.
If the Object is already on the Correct Layer then the only way to do this is to Select By Layer and Change the Properties of all the Objects to the Name of the Layer - You will have to type in the Layer Name.
The Standardize Layers command is the easiest way to change the names of objects to do what you want here.
Alan