The torn off button seems to be an important puzzle piece throughout The Next Three Days' plot line. As I understood, it was the missing evidence that could've granted freedom to Lara Brennan. However, it was never found until the very end where we see it unnoticeably falling into the sewer when detective Quinn tries his luck searching for it with his bare hands. So what was the importance of showing that scene to the viewers? In the end, Lara remained guilty in the face of law.
It confirms the protagonists' actions from a moral viewpoint.
First of all, the button's existence (as well as the flashback of the murder) actually clarifies that Lara is not guilty. While the Brennan family are the heroes of the movie, it was never entirely clear if Lara might not actually have commited the murder. While the primary focus of the story was on getting her out of prison and even John might not have cared for it completely, there was always some uncertainty and ambiguity about her possible guilt resonating in the background, which also put somewhat of a moral questionability to John's plan and the whole film's premise.
This ambiguity is also something that the filmmakers deliberately included as a divergence from the original version, as expressed by writer/director Paul Haggis in an interview with Script:
They made it quite clear from the beginning of the film, she was innocent, and that he was loving, and he’d do anything to get her out, and, in the end, they lived pretty much happily ever after. The bumps along the way were good but I thought I could make him pay a larger price...Would you save the woman you loved if you knew that by doing so you’d become someone she’d no longer love? That interested me. And that wasn’t in the French film at all. [...] The whole issue of innocence was fascinating to me because I didn’t necessarily want to say whether she was guilty or innocent. I just wanted John to be the only one who believes she’s innocent. The evidence is overwhelming. [...] In the French film, they’re coming home from dinner—they’re loving, they’re kissing, you’re starting to think, “Oh, this woman’s innocent.” You just know from the beginning. I wanted a scene in which you see that she has a temper by creating the Erit character who’d get under any woman’s skin. So, you later think back and say, “Oh, she almost clocked that girl. She could have gotten into a fight. She could have done something. She could have hit her boss.” So you have doubt.
So the flashback confirming Lara's story, especially the button she kept insisting existed, resolved that ambiguity, albeit only at the very end, and pracitcally confirmed the Brennan family's actions in a moral sense.
However, you are right that she still remains guilty in the view of the law. And I think that had a similar purpose. The cop missing the button (and the implication that the law probably won't change its view of Lara's guilt) reinforces that the Brennans did the right thing and succeeded where the system failed. It puts a certain kind of definiteness to their decision and their escape when it confirms that there is really no turning back. Lara will stay a convicted murderer on the run and fleeing the country was their only real chance.