I have seen both Outlaw Star and Firefly and love both of them. I definitely thought of Melfina in Outlaw Star when I was first introduced to River in box. I also noticed some similarities to Cowboy Bebop as others have brought up. However, even if Firefly was influenced by these, it does not mean it was a direct rip-off. However, just because I have seen this conversation all over the internet for years, I thought I would throw in another precursor sci-fi set in a very similar world to Firefly, Cowboy Bebop, and Outlaw Star that I never see mentioned. Blake's 7, a British sci-fi from the late 70s created by Terry Nation who invented the Daleks of the Doctor Who universe.
In Blake's 7, Blake is a freedom fighter who has taken a stand against the evil Federation, a earth-centered government that bullies the outer colony planets. He was previously a leader in a group of rebels that banded together from various outer rim colonies to resist the earth-central government and was beaten. He now has a collection of rogues, hackers, con-men and other outlaws that ride around with him in a special ship, with a personality, that interacts with the crew and is basically considered a member of the crew *cough* Gilliam. Also some of them are quite willing to back stab each other, abandon each other, or even consider killing each other if the cause or the price is right much like Faye Valentine, Aisha Clan Clan, or Jayne Cobb. They are pursued by a government agent who is intent on apprehending or killing them and stopping Blake from spreading his message which would encourage further insurrection towards the corrupt earth central government (The Operative in Serenity). All that said, there is no analogy to Melfina or River in this show. However, there are a lot of striking similarities.
Do I think that just because it predates each of these other shows, they all ripped off Blake's 7? No, but I believe they were in some way influenced, even if not directly, by the story-telling and the world. Ideas influence other ideas. People love the Matrix, but without Ghost in the Shell, Jon Woo's bullet operas, and Blade Runner there is no Matrix. However, the Matrix is not any of those movies. It was influenced by each of them though. However, the Wachowski brothers freely admit these influences, even pointing out that they tried to recreate some scenes from Ghost in the Shell. Melfina's introduction and River's introduction are a bit too similar, it is almost hard to believe that no one working on the show hadn't seen Outlaw Star given the similar way the character is introduced... but that said the show in its entirety is no way a rip-off of Outlaw Star anymore than Outlaw Star is a rip-off of Blake's 7. In fact aside from Melfina and River, there are probably more similarities between Blake's 7 and Firefly, but still I don't think it's a rip-off. We see similarities in that they all tell similar stories that we like and/or connect with. They all have a semi-optimistic David vs Goliath aspect along with a dark view that the system is always stacked against you and that the future is not a safe or happy place to be all of the time. We haven't solved all of our problems and we must fight for freedom against a massive and oppressive force. They are all great shows in their own right and in their own times... so let's just enjoy them.