Any social change movement, by definition, begins by trying to persuade a majority of people to care about something they have usually accepted as a fundamental of the world. Worse, it must move a majority of people to an active rejection of the status quo.
So what do you do o change opposition? Because to succeed, you are going to have to change people's minds.
Sometimes, you can't. The worst case scenario on this was the Civil War and the "intractable" problem of slavery. The choice presented to society was stark, there were no half-measures, and the issue was so fundamental to the definition of society that there was no way to resolve it short of armed conflict where one side beat the crap out of the other. Even then, completing the social change necessary to eliminate explicit discrimination as a matter of law took another 100 years. It is safe to say we are still working through the issues.
Ideally, you want to avoid this kind of worst-case scenario and see if there are other ways to persuade a sufficient number of people that your social change is worth it. Which raises the question of how.
First, you need to triage. You are never going to get people to change their minds all at once. But it's important to think about who is against you and why. There is a difference between a reflexive "ick," an objection based on economic motivation, an objection based on fear of displacement/incumbency, an objection based on religious principle, or an objection based on other societal values. there are also differences of degree.
For example, consider the question of "immigration reform." You have a substantial opposition to any reform that would either increase the number of immigrants entering the country legally and/or to any "amnesty"/"pathway to citizenship" for those already here illegally. But the reasons for this position vary significantly. For some it is fear of displacement/fear of change ("I'm losing my country to these foreigners"), for others it is economic ("these people are taking my job and driving wages down. I used to make good money as carpenter, but now the contractors just hire day laborers") and for others it is ideas about fundamental fairness ("these people broke the law, we cannot reward that because it is unfair to me/my parents/grandparents who obeyed the law"). Different arguments, and different solutions, are likely to prove acceptable to different groups.
For example, those concerned with fundamental fairness are more open to the idea of something like "The Dream Act" which would grant citizenship to anyone brought into the country illegally as a minor and who served in the U.S. Armed forces. Most people understand that when you are five, you really don't have much sy in the matter if your parent(s) bring you here, and the willingness to perform service for the country alters the "fairness" calculation. But such an approach is unacceptable to those concerned about changing the character of the country, since these are still "foreigners" rather than "real Americans" (i.e., people like me).
Which brings us to the subject of the previous post. Who is your opposition on same sex marriage, and how do you change enough of them? Remember, you are starting in the hole. If you want the change, you must persuade a sufficient number of people to agree to the change. This is not a question of "should" but of what is. Similarly, because you are targeting opposition, you have a limited opportunity to frame the debate. At the same time, framing the debate is a key component of persuasion. If the other side successfully frame the debate, you usually end up losing. But at the same time, you need to engage with people in a way that is relevant to them.
For example, the women's reproductive rights movement has attempted to keep the same frame for the debate for the last 60 or so years: women's rights. The right of any individual woman to control over her body. The anti-reproductive rights movement has used the same frame for about 35 years now -- abortion is murder of an unborn child. The anti-abortion activists have had the upper-hand for some time now because they have successfully framed this in a way that makes the abortion issue appear irrelevant to their lives despite the frame of reproductive rights advocates. the anti-abortion movement has succeeded in making it possible for large numbers of women to view the question of abortion as utterly separate from the overall question of women's reproductive freedom, personal choice, and equality. Telling these people that they are anti-women's rights fits the pro-choice frame, but is unpersuasive at this point unless you can do something else.
[As an aside, this is also why the attacks on access to birth control enjoy much less support, even within generally "pro-life" communities. Pro-choie activists view these as different flavors of the same issue. But a sizable portion of the country has no problem distinguishing between birth control (autonomy/freedom/empowerment/privacy) and abortion.)
Which again circles us back to same sex marriage. Obviously, if you are in favor, you want the frame to be a civil rights frame, a fundamental fairness frame, or a privacy frame. For opposition, the frame is either a religious values frame or a fear of disruption/change is bad frame. (To distinguish: 'God says homosexuality is an abomination' is a religious values argument. 'Your kids will become gay and you won't have any grandchildren unless they adopt or use a turkey baster' is a fear of disruption argument.) Successful arguments need to engage in a way that is meaningful while simultaneously moving toward your preferred frame.
This is not easy. In fact, this is one of the most difficult areas of successful advocacy.