I think this question can actually be interpreted in two different ways, since there are essentially two possible courses to give:

1. One whose purpose is to enrich the students' lives (as Ilya Grigoriev put it in his comment) - such a course would probably do best to adapt ideas mainly from Noah Snyder's answer - scales, statistics (probabilistic thinking), and I'd add a few other topics (maybe small logical puzzles to make students see how "thinking mathematically" may have a positive effect on their analysis of every day situations).
2. A course whose main purpose is to convince students that "math is great". Make them drop the false impressions they've been fed with their entire lives, about mathematics being a "dead science" ("haven't all math problems already been solved?") - this can be achieved through looking at mathematics from a historical perspective, especially putting emphasis on problems solved recently (say last 50 years), open problems and new emerging fields in mathematics. Making them see that math can be fun is perhaps the ultimate goal, as Kevin O'Bryant mentioned.

Now the question arises: which of the two courses should we teach? Morally, if we think of the benefit of our students, we'd have to pick the first. But if we are mainly interested in "advertising" (which I don't think is a bad idea!), we should pick the second. Perhaps if such an "advertising course" is sufficiently good, it would convince them to take a second course, more along the lines of #1 above?

A compromise could be to divide the course into two parts - after we've convinced the students that math can be cool, you can go on and teach them more traditional stuff that will be beneficial to them.