I met John Lilly last October on a decidedly odd techie scavenger hunt in Hawaii–trapped in a van all day while searching for tikis and such, as if we were a geek Brady Bunch (yes, BoomTown works that hard for you)–and found him to be just the kind of quietly intelligent, thoughtful and self-effacing digital exec that Silicon Valley could use a lot more of in these often frivolous, look-at-me Web 2.0 days.

Soon after, in January, he was named CEO of Mozilla Corporation, after serving as its COO. Lilly took over from Mitchell Baker, who remains Mozilla’s chairman.

The open-source software nonprofit foundation, also a for-profit start-up, is most famous for its red dragon logo and, more importantly, its increasingly popular Firefox browser, whose share has steadily increased since it debuted in late 2004.

Here’s my video interview with him about what’s next:

Firefox now has about an 18-percent share, which has climbed from 11 percent just two years ago–against Microsoft’s (MSFT) Internet Explorer browser juggernaut with 74 percent and Apple’s (AAPL) Safari browser with 6 percent.

Mozilla’s recent launch of Firefox 3 should help it grow further, even though IE 8 will soon emerge out of beta too.

But on its record-setting debut day in mid-June, there were 8.3 million downloads of Firefox 3 in 24 hours. Current overall downloads have just moved past 50 million.

That means more money in the bank, although mostly from Google (GOOG), which pays Mozilla royalties for Google ad clicks that come from Firefox searches (Google is the default search box on the browser).

All is not sunshine and daisies, of course.

Besides having to keep up the innovation, Mozilla also has to keep up its technical prowess–despite the fact that it is not likely to go public and shower employees with giant gobs of overhyped stock, given its unusual status as both a profit and nonprofit.

In addition, in case you haven’t noticed, the start-up game is still rocking and it remains awfully enticing to jump into the ever-crowded pool.

For example, Mozilla lost Engineering VP Mike Schroepfer last week to Facebook, where he will be the social-networking site’s director of engineering.

But Lilly took that departure calmly, wishing him well in a classy way on his own blog, just as he seems to do most things.