I think that the Civil Rights actions taken in the 1950s and 1960s might be one good example of a dissent group trying to make a law. I would also suggest that the National Organization of Women (NOW) successfully advocate for the passage of the Violence Against Women Act (VAWA). This particular piece of legislation helped to make domestic violence no longer a private issue, but rather one of public safety. Passage of this legislation helped to shine a much needed light on the issue of violence against women in the realm of the private, ensuring that spousal abuse became an issue that resonated in both the political and social consciences of the nation. This is an example of a dissent gorup having influence to help to make a law.
If you think hard enough, you can come up with tons of groups in America that would like to change one or more laws. Recently, there has been more and more talk about the legalization of marijuana. The arguments for the legalization of marijuana are many. For example, the legalization of marijuana would lead to less drug wars, increased revenue for the government in terms of taxes, and more law enforcement concentration on other areas of importance. I am not in favor of the legalization of marijuana, but many dissent groups in America are and they seek change.
Expanding on one of pohnpei's points, there was an event yesterday where President Obama was in California attending a fundraiser/rally for Senator Barbara Boxer, a Democrat who is up for re-election, and some people in the crowd started yelling at the stage to repeal the military's "Don't ask, don't tell" policy against gays openly serving as homosexuals in the military.
The gay rights movement campaigned for Obama, but also made him promise to repeal the policy once in the White House, which he did. He has already relaxed the enforcement of the policy, and this group was asking them to do more.
The most obvious example of dissenting groups getting laws changed is the Civil Rights Movement of the 1950s and 1960s. Through their protests, these dissenters managed to end the system of segregation that had once existed in the United States.
There are many more current examples of groups that try to change laws. Perhaps the most well known of these groups right now would be right to life groups. These groups are pushing to get the Roe v. Wade decision overturned.
You can also mention groups on both sides of the gay marriage debate who are trying to change laws on that subject.