To make a compelling argument either verbally or in writing requires tact, knowledge and the ability to see both sides of the debate. The credibility of a compelling argument can be the difference between winning the debate or argument and being discredited.
Educate yourself on the subject matter. If you don't know the details of your argument, the opposing side will find holes and weaknesses in your argument which can lead to an unsuccessful debate. Study not only your side of the argument but any potential opposition as well.
Have a passion for your subject matter. Being passionate or expressing belief in your argument will help express the validity of your side of the debate. As with many debates and arguments, believing in your subject helps, but it's not necessarily needed to win the debate.
Don't let anger rule the argument. Getting angry or flustered greatly reduces the chance of making a compelling argument. Step back and look at the subject matter to find any weaknesses in the opposing party's argument. Trying to make an opponent angry in a debate or argument is actually a tactic used to make an opponent make mistakes and lose credibility.
Listen as well as talk. Great speakers are usually great listeners. Listening to and acknowledging an opposing side in an argument shows that you do understand the opposing argument, but have the conviction to stand by your opinion.
Back up your argument with statistics or facts. Having an opinion on a subject matter, passionate or not does not necessarily lend credibility in making a compelling argument. Have statistics and cite sources when available to back up your side of the debate.
Practice alone before the debate if at all possible. Practicing responses and statements will help you deal with questions and remain calm. If the argument is done in writing, read it aloud to see how the subject matter flows and what message is being sent.