Open the top level assembly and do a pack and go. In the pack and go you can change the save as name for all the child parts of the assembly (including top level assembly itself) in the same window. This will not only save time in opening each file and doing a save as but will also keep the references linked to the assembly file.
You can also do references through the Save As dialog. Like Deepak said, start with your top-level assembly or top-level assembly drawing.
There is a button in the dialog that allows you to change the name of the other files that are referenced in the assembly. Be aware if there are any drawings connected to the referenced parts they will not go along with it so you would have to reattach the part/assembly files to their appropriate drawings and then save the drawings. Also know that when changing the names in the dialog, you have to select off of the field in order for the change to take place. When you click off the cell you are changing, it will turn green to highlight the item that were changed. If you haven't done that the change doesn't happen.
The difference between the two methods is with a Pack & Go, the file you have open is unaffected and with the Save As method I described, the files you have opened become the new parts you saved. With Pack & Go there is also an option to include the drawing associated with all part/assembly files, so the drawings can go along for the ride (this is helpful is you use Start Parts with drawings already started).
Great! I use the pack and go as we make like systems. I didn't consider it for this operation. Thanks.
Both answers , Deepak and Matt are correct. Either option, depends on user.