Thanks for your recommendation. I believe this is the answer I was looking for. I found it on a google search.
Action: Rename
New Name: "Table1_" & Format$(Date, "MMDD"

Object Type: "Table"
Old Name: "Table1"
Save the macro, then run it. Notice the funky table name?!?
Now to fix this, click on the new Macro, then choose from the Access menu:
Tools | Macro | Convert Macros to Visual Basic
This creates a new hunk of code in your Modules called "Converted Macro- Macro1".
Double-click on the new code. Don't worry, it won't automatically run the code!
Then edit the code and take out the quote marks until it looks like the line of code above...
Option Compare Database
Option Explicit
'------------------------------------------------------------
' Macro1
'------------------------------------------------------------
Function Macro1()
DoCmd.Rename """Table1_"" & Format$(Date, ""MMDD""

", acTable, "Table1"
End Function
Finally, change your original macro to the following:
Action: RunCode
Function Name: Macro1()
Then, when you double-click your original macro, it will call the piece of code.
I just double-clicked it and my table was renamed to "Table1_0205".