The Rename menu comes with a command for adding the current date/time to names:
Finding the latest file below a folder and using its date would also be possible, with a bit of scripting.
Custom Column - Newest File - Directory Opus Resource Centre contains some simple VBScript code showing how to get the most recent file below a folder, which could then be used in a Rename script to add the file's date/time. If you need detailed help with that, please link your account.