The title of this post was a popular search string for this blog yesterday. Since I like to try to make people happy, here's my attempt:
1. If the person is politically conservative, point out that if they believe in small government, either every adult or no adult should be allowed to legally marry. Any other option just creates more bureaucracy.
2. If the person is fiscally conservative, point out that more people marrying provides income for the government from marriage licenses and boosts the economy with increased spending on the marriage industry: gifts, parties, honeymoons, etc.
3. If the person is religiously conservative, point out that a civil service is not a religious service. The First Amendment protects the rights of churches to provide marriages for whoever they please. A couple of years ago, a Kentucky church banned interracial marriages, and there was no legal challenge, because they have the right to define religious marriage as they wish.
The best argument I know is an old joke. I'm sorry I don't remember the comic or the original version, but I tend to tell it like this: "Gay marriage is not a threat to my marriage. Salma Hayek is a threat to my marriage."
ETA: While it probably wouldn't change their minds, you could also mention this fact: Divorce Rates Higher in States with Gay Marriage Bans.