Suppose Splnter of the Minds Eye was a big hit ,

A noticeable spike in mud pit fighting. 

This is a VERY interesting question! To answer it I have to look at the lore created by 'Splinter of the Mind's Eye'. And I would say that it would've radically changed the future of SW.

As Jan El Señor said, Luke and Leia would not have been siblings. And Vader would not have been their father. But I think the impetus goes beyond that. In 'Splinter' Leia was more like Saw Gerrera and less like Padmé. She was more of a rebel and less of a diplomat. Luke is more self developed when it comes to the Force. And in his duel with Vader he manages to triumph over Vader; slicing his arm off and leaving him for dead. Also the mythos about lightsaber skills is different with 'Splinter'. Throughout the OT and the PT the mythos of the lightsaber is that it can only be handled clumsily by non-Force users; and you need a significant amount of training with it. But in 'Splinter' Leia uses Luke's lightsaber to duel Vader and buy time until Luke was freed from being pinned.

What's interesting is that the mythos of Luke, Leia, and lightsabers in 'Splinter' matches up much truer and better with the current mythos of the Disney sequel trilogy.

Where would it go after 'Splinter' if it were a blockbuster? I think it could've gone in 2 directions. They could've killed off Vader and made the emperor the new and more powerful baddie that Luke would have to defeat in the final chapter of the trilogy. Perhaps Leia continued to develop her fighting skills and helps him. But another (and more likely direction IMO) is that Vader gets a new arm and is motivated by revenge to destroy Luke. Also Han Solo and Chewbacca were not in 'Splinter'. So there would be less side plots. I think in the third act of the trilogy the duel between Luke and Vader is more centered around the Force. Luke ultimately wins this time because he learns the lesson Yoda taught him in TESB: the Darkside is not really stronger. It's just quicker, easier, more seductive and Luke at some point is tempted by it.

If the sequel were to be highly successful then I think it further changes things. The prequels may have never been made. Or if they were it would have a totally different story since the story of how Luke's father became Darth Vader is no longer on the table.

Maybe Lucas retires SW after that. Maybe Lucas decides to make the sequel trilogy first instead of the prequels. Or maybe he sells to Disney after the 'Splinter' sequel and the current trilogy gets made mostly the same way; matching up better with the 'Splinter' version of the OT. But I don't know if Han or Chewbacca would be a part of it since you would be bringing them back all the way from the first movie. And maybe Kylo is not Han's son or even Leia's. But maybe he turns out to be Vader's son or grandson. Hard to say.

Anyway I think that's how things would've changed.

Han Solo at Stars End (1983)

Han Solo's Revenge (1986)

Han Solo and the Lost Legacy (1989)

Splinter of the Mind's Eye Awakens (2001)

The Last Splinter of the Mind's Eye (2004)

Rise of Splinter of the Mind's Eye (2007)