-The main thing we'd keep from the Firefly aesthetic is the "keeping the ship running with duct tape and prayers" part. With a bit more emphasis on the prayers. Everything else is up in the air.
-There's still a touch of the outlaw in this family, because there's got to be a reason they choose to live this nomadic life in space rather than settling onto one planet.
-Religious persecution is the too-easy answer. I don't envision this future society being super religious, but I also don't think they're actively hostile to it, because I want religion to be part of the background, not the focus.
-Maybe some variation of the Firefly "we were on the losing side of a rebellion" thing?
-I also think it could be fun to play with my long-running idea of a society of spaceship-dwelling "Drifters", and combine it with some inspiration from the 1890s.
-I keep thinking of Laura Ingalls Wilder taking a covered wagon to Missouri, but being absolutely adamant that, "We are not wagon people." With the implication that "wagon people" are an unsavory class of people.
-Drifters are "wagon people". They live in space and travel around looking for work on various planets, never sticking around long before moving on. Other people, both planet-dwellers and travelers, distrust them and look down on them. Among themselves, they have a pretty vibrant subculture, sharing traditions and helping each other out.
-A few decades ago, there was a boom of migration as people moved out to newly-terraformed planets with incentive from the government. Now, lots of these planets are facing ecological and economic problems, leading to a lot of people traveling to other planets looking for better opportunities elsewhere.
-I'm not sure if this is the origin of the Drifters--some people just never settle down, and form their own spaceship-based subculture--or if the Drifters have existed a long time, and now have to deal with this new influx of travelers who could benefit from their help, but don't trust them.
-It could be fun to translate a "covered wagon" set-up into a spaceship. A living/transport capsule is a fairly simple, cheap, accessible technology that your average settler can afford, and they have to buy/rent/borrow an engine set-up that they can attach the capsule to. Maybe some engine constructs are able to carry multiple capsules, setting up a wagon train situation.
-The main action would take place on one family's ship. There's a married couple that owns the ship, several grown-up children who help run it, some of whom have spouses and children of their own. There's at least one grandparent and a couple eccentric aunts/uncles/cousins.
-The father and mother are excellent captains and mechanics who can keep their ship running with duct tape and paper clips. At least one of them has some kind of surprising military experience.
-One in-law is a young man from a well-to-do family who moved out to the terraformed planets and fell in love with a daughter of the family, who made it a condition of the marriage that they had to stay with her family's ship. He found this acceptable and fits in well, but he's got some more polished habits that are a bit of an odd fit, and he's still learning about Drifter culture.
-Maybe this family takes on a passenger who's secretly a gunslinger running from his past. He thinks he's going to have to save them, but they wind up saving him and giving him a place to belong.
-Or maybe they sometimes travel with a priest who has a past as a gunslinger.
-(You say "western", the only thing my brain gives me is "gunslinger". Which may be a reason I'm not well-suited to actually write this.)
-That, and the fact that this is all very vague aesthetic rather than anything resembling a story, but it's fun to play around with it.
