This is a sensitive topic, but I intend my words in a constructive and supportive tone. I hope they will be accepted in that manner.
I have quite a bit of experience at being the most conservative in the room. As an undergrad, I was at a secular university which had little patience for evangelical theology. One of the theological schools I attended (the one from which I received my MDiv) is in the running for the most liberal in the country; most of the issues here pale in comparison to what I faced there every day. My denomination is incredibly diverse, with a disproportionate number of the clergy being liberal. Therefore, I've had to learn some skills not only for survival in environments where I am the theological minority, but also techniques which help my convictions to be heard in a productive way. This hasn't been easy knowledge for me to learn, so I hope it is of value if I pass on a few pointers:
Beware of falling into the "us verses them" trap. It's so easy to feel like people are against us, but it can lead to a counterproductive defensiveness or even aggression. That doesn't win anybody over, and just raises everyone's blood pressure.
Instead, I've found it beneficial to recognize when people are working towards the same values as me, but interpret those values in different ways. For example, I may believe that one course of action is a just one, and others may believe that another is more just. We disagree, but we share the principle of justice. This can also apply to other values, such as holiness or compassion. Being able to say to one another "you and I are both seeking ___ in this situation" can lead to a mutual team effort to find a solution, rather than further polarization. Furthermore, people are more likely to listen to us when we acknowledge the positive values for which they are working.
Which leads me to: Listen. A useful technique for me is to paraphrase back what someone is telling me and ask "is this a correct understanding?" I try to get to the point where I could articulate the other person's position, and their reasoning, even if I do not agree with it. This helps me to understand where they are coming from and to be alert to possible solutions to a conflict. I can't recommend this simple approach highly enough. Combine that with being willing to refine your paraphrase according to the person's feedback, and you'll find communication gets a lot easier.
Be cautious about ascribing motives to someone else. We can't see into someone's hearts or minds, and human communication is imperfect -- so it's all too easy for us to misinterpret someone else's motivations. Without strong evidence to the contrary, I find it's generally best to interact with people using the functional assumption that they are representing their motives accurately. Doing anything else is insulting to the other party and can cause communication to deteriorate.
Related to the above: Avoid character judgments. People don't like that and will tune you out.
Assess what is essential and what is non-negotiable. Determine if there is an area in which you can be flexible and find room for compromise. There are certainly some thing that cannot be compromised, but not every issue will fall into that category.
Last but certainly not least, remain calm. Present your position clearly and methodically, detailing the values you are trying to pursue and why you think your course of action is the best for the purpose. If you feel like you might lose your temper, take a step back until you can continue on an even keel. If attacked, do not respond in kind, but maintain respectful and rational discourse. You represent your position best when you yourself remain collected.
As I said, these are some lessons that I have learned over the years being "the token conservative." I would like to think that the growing pains associated with these lessons didn't go to waste, so I hope it is useful for me to pass these tips on.
Mac