If you mean the characters in the logo (as opposed to the background behind the logo) turn from red to green, then you need one of the tft type display panels. The nokia 5150 (single color) lcd driver has a graphics library that can display graphics, I expect the library for the color display listed does the same (although you should check that before buying one 
) if not it is possible but more complex to write a bit map in to the controller to form the logo. As the logo is a graphic being displayed on the panel, it can be changed (with varying amounts of difficulty and brightness) from red to green. Brightness may be an issue as well, leds would be brighter than the tft display. You would have to experiment to see what works for you (and what works in your Arduino, you may have to upgrade the processor to a mega or an arm to get enough memory and I/O pins to do what you need).
(edit: ) Note that most of the tft displays are 3.3V and thus need level translators to run from a 5V Arduino (and you will likely blow up the display if you don't use the level translators) so if you decide to go this route ask about level translators before doing the hookup.
Peter