Mongoose Traveller (and Classic Traveller, on which it was based) use a relatively rules-light system, with most tasks (combat or otherwise) resolved by rolling 2d6 plus/minus modifiers for skills, attributes, difficulty, conditions etc; a roll of 8 or more is a success. Character generation is almost a game unto itself and resolves your character's previous career - unlike D&D, you very rarely start at the age of 18 in Traveller - you start as an older guy with serious professional experience under his belt. Classic Traveller chargen is FAST once you get to know it well, about 5 minutes (!) on average; Mongoose Traveller chargen takes somewhat longer because it is more detailed, but is still quite fast.
The great thing about Traveller is that it has A LOT of rules for starships - ship design, ship combat, cargo capacity, interstellar trade, passenger services, you name it... It also has good and fast rules for detailing worlds and/or generating worlds randomly. The tech level is also perfect for Firefly, with emphasis on low-tech slugthrowers (including shotguns and revolvers... IN SPAAAACE!) rather than energy weapons and without too many cybermods, biomods or robots milling around. It also has rules for psionics if you want to model River Tam or the Agents.
Traveller can be used to run a very wide variety of sci-fi campaigns, from mercenaries.military to trade to exploration. That said, the default campaign is having a tramp trader and going between the stars doing speculative trade, smuggling, odd jobs, adventures and passenger transport. Just like Firefly.
Also, if you use 15mm miniatures like Traveller suggests, you can get a lot of Firefly-like minis from Ground Zero Games and Splintered Light Miniatures.