To many advocates of extending the Space Shuttle, the most important thing is jobs. They'll tell you about how the crew at the United Space Alliance are doing such a fantastic job and it would be a shame to lose this national asset. Which is funny because, back in 2004 when the Shuttle retirement decision was made, we were told that the workforce is aging, retiring, and unable to get new blood.
However, a rare few who advocate Shuttle extension do so because they strongly appreciate the capability of the Shuttle, and cite Hubble servicing missions as the greatest demonstration of these. I, for one, think this capability can be better served by more cost effective means: building satellites so they are roboticly serviceable to begin with, or the robotic/manned options for Dragon presented by Max Vozoff in March.
But it is interesting to think about how one might utilize the orbiters once they retire. The current plans are to ship them to museums and stare at them in wonder.. that seems like a shame. Richard Godwin and Dennis Wingo have suggested that the orbiters could be "retired on-orbit". Maintaining their capability. They are undeterred by the long list of retrofits that would be required to make this workable, and maybe even a little naive, but I have to give them credit because they actually recognize the problems of the Shuttle's on-orbit endurance.