the reason for this "duplicate button press" issue with the dying battery is that the boxee remote sends out two different signals when you press a button - one indicating a good battery level + the key you pressed... one indicating a low battery + the key you pressed . This is so the boxee UI can show the Low battery Indicator on the main menu.
The problem is that when the battery is on the border between OK and LOW, the remote goes mental and sends out BOTH signals when you push a button - boxee then recieves the key you pressed signal many times.
Boxee Box > HDMI > HDTV
2 x HTPC (boxee 0.9) > HDMI > HDTV
2 x 8TB NAS