Is it just a naming convention? Is there a rule in the English language that dictates the order they are listed? Was one of the two more important than the other? What about other duos? Who decides the order of their names?
The simplest answer is that Lewis was a US Army Captain, and designated commander of the expedition. Clark was a Second Lieutenant (few ranks lower) and essentially a close friend, brought in by Lewis himself, so they were not equals in some minor ways. These two were the only commissioned officers however.
In 2015/2016, this is much more of a linguistics question than a history question, at least compared to how it may have been in 1804. Not to say there's not a historical basis, but just that it wasn't necessarily the case that we would still say it this way, as the order of such phrases can and do change over time. So I'm going to address some of the the linguistic factors behind it.
This is meant to supplement responses strictly on the historical origin of the order.
It may be the case that because Lewis outranked Clark his name is first as /u/mycd said. This is normal when it comes to people where there's a hierarchy involved. But today the reason you say it in that order is because everyone says it in that order. You could have also asked about why we say "Peanut Butter and Jelly" in that order and, linguistically, it's the same question.
It has frozen in the language in this order. This is called a colocation (among other names, such as a freeze). For speakers of the language, "Clark and Lewis" will not sound right, any more than "McCartney and Lennon" or "Clyde and Bonnie". In those cases it's not a matter of importance.
However it's also worth pointing out that we might not always say Lewis and Clark. We might say Clark and Lewis some day
You might also be interested in Panini's Law which governs the order of many of these types of colocations, but again with room to move around for things sounding more catchy or rhyming better in another order.