I hesitate to share this, because some people have freaked out when we tell them how we do stats. But it's worked for us for years.
We let people create their character concept, and allow them to choose stats that they think are appropriate. No rolling of dice, we just let players assign whatever values they think are appropriate for the PC they've created.
Interestingly, no one in our groups has tried to take advantage of this. It's very rare to see anything below 10, but it's also very rare to see an 18. We tell them to think very seriously about what their scores should be, based on their concept, and they're fair.
I wouldn't try this with a completely random group at a gaming store, or with young kids, but we get ability scores that are generally in the same vein as "roll 5d6, choose the three highest" and in many cases 4d6 choose 3.