I should probably tip my hand to start and say that I don't know how to change a mind. However, I have convinced some people to be vegans, and have at least gotten a substantial other amount of people to rethink their obligations to animals. Part of the problem in advocacy is that everyone has their own unique mind, and they might all require different strategies or arguments to convince them one way or another.
One helpful tip is to know your audience. Which I generally hate as a "helpful tip," because most people do know their audience, and we are constantly adjusting the way we talk based on our audience without even realizing it. Despite this, it does help to break down certain potential pitfalls in conversation, especially with members of particular groups, as I addressed in my post The Argument from "Marginal Cases".
As my friend pointed out in the comments of that post, there can often be a lot of resistance to veganism and animal rights from those involved in other advocacy movements. In one sense, this isn't at all surprising. The thing about unjust privilege is that it makes it hard for you to realize that you're a beneficiary; it appears to the privileged as if it is deserved.
But in another way, one might accept that this group would be more receptive, because they're used to understanding and breaking down structures of privilege. To some degree this is true; the advocates for veganism I know often have involvement, or at least interest in other movements, and I wouldn't be surprised if members from those movements were on average more receptive to veganism. But all too often, advocates against other forms of prejudice resist any notion that speciesism is meaningfully analogous to sexism, racism, etc.
One especially troubled line of argument, I think, is analogies to particular atrocities, like slavery or the holocaust. These analogies, no matter the context, are often fraught with more complications than they are worth. You see this in discussion of civil rights for LGBT citizens; the analogy towards racial civil equality is obviously at least partially apt, but I think it often derails the conversation. Opponents of equality will offer many differences between these different forms of oppression, all of which may be accurate, but this hardly moves the conversation forward.
Most dreaded of all is the thought that one might be claiming that some tragedy is "as bad as" another. I've often heard that the death of a chicken cannot even compare to the murder of a human. Perhaps not! I'm inclined to think that the two can be compared, but my argument doesn't really depend on that. It's really not relevant. All I need suggest is that killing a chicken is wrong, and thus we should refrain from doing it. If you want to draw up a chart saying how bad each kind of animal death is compared to humans, and how bad each person's death is compared to every other person's, feel free. It's sounds like a terribly depressing exercise and I'm not sure what it would accomplish.
Instead, I prefer to think of our relations to others, and how these relationships shape our obligation. I think the most fundamental relationship is the recognition of someone else as an other, as someone who has claims to things and reasons of acting of their own. The most fundamental claim an individual can have is the claim on their own life, and their right to live it.
There are many ways our rights are infringed upon, and different forms of oppression may take analogous forms. Since each of these forms is reprehensible in it's own way, in can seem tacky or degrading to draw on these analogies to make a point, even if the point is apt. I think it's often best then, to avoid drawing analogies (when we can) and get people to think through the fundamental relationship with others, and reflect on what obligations are implied.