In a society that is just barely holding on by an unraveling thread, he wants to maintain the rule of law and preserve democracy. He wants everyone to remember who and what they were before the Cylons tried to wipe humanity from existence.
THIS. This is the fundamental reason why I want to flick people who bitch about Lee as a character right between the eyeballs. Despite having his own baggage, he doesn't have quite the edgy, ragey baggage that some other characters on the show have but that doesn't mean he isn't an interesting character, and a really really valuable character. RDM rightly notes in the roundtable podcast that as the strong-willed idealist, Lee is the hardest character to play on the show, and obviously he realizes that in the many plotlines they sought to explore and develop crowded the playing field such that it only made it harder, both for Lee's character arc and for Jamie. Which makes his performance that much more wonderful.
Also, it's worth pointing out that this aspect of his character was there FROM THE BEGINNING, and is a perfect starting point for understanding why taking him out of the cockpit and putting him into the Quorum was less out of left field than it was made out to be. It's downright logical, in a sense, if you contrast it with the bits of his "Crossroads Pt. 2" monologue where he goes on about the fleet making their own laws and justice now. Did Lee's political life really get a chance to flourish in the show? No. I think we really needed five seasons for that to happen, so that we could see him take his rightful place as Laura's protege. (I don't ship Lee/Laura, nor Lee/Kara - I don't really ship anybody, but I deeply wish we would have had a more uniform, well-constructed arc where he started out with this fundamental mentor relationship, whereby she presented him with an intriguing, alternate philosophy and career path to his father's, yet he stuck with the uniform for a long time until circumstances eventually put him where he was really best suited all along.)