These strategies can increase your likelihood of success when negotiating.
Develop a rapport with representatives of the publisher. A 2004 experiment, whose results were published in the Harvard Negotiation Law Review, found that subjects were four times more likely to reach a mutually beneficial agreement AND feel good about the process if they engaged in five minutes of small talk over the phone before participating in a negotiation simulation via email.
Negotiate for the win-win. When both you and the publisher get something you need, success rates go up. Try not to approach a negotiation as an adversarial situation where you win and they lose. Both parties have the same goal: to get your work published.
Always ask (early and often!) If you don't ask for what you need, how can you expect to get it? Don't assume contractual matters are written in stone. Generally you should wait until your article has been accepted before negotiating. However, you can always ask for a copy of the standard contract ahead of time so you can plan your negotiation.
Know your alternatives. What is your next best option if negotiations fail? Knowing that should give you confidence and strengthen your hand going into negotiations.
Pick your battles - how much do you care? Not everything is worth fighting for, and a few important wins are more likely that many less important ones. For example, you may want to spend more time negotiating author agreements for original, high-impact research as opposed to a minor op-ed or book review.
Make a practical case for the publisher to say yes. Show them how it is to their advantage, or at least not to their disadvantage, for you to get what you want. Problem clauses may have been put into contracts years ago, and publishers may no longer care.
Be a good partner. Cultivating relationships with publishers over time leads to better outcomes. Success once makes a precedent, meaning that you'll likely get what you want again next time you work with that publisher. Faculty editors can be powerful advocates on your behalf. Cultivate strong working relationships with them.