Lucasfilm has now officially confirmed that Woody Harrelson will be joining the Han Solo movie set for release in 2018. While the official announcement said nothing about what character Harrelson would be playing, the initial rumors that came out before Disney spoke publicly said that the role would be that of the mentor to Han Solo. Since the rumors were correct about the casting, we can assume they were probably right about the role as well. As it turns out, there's already a character in the expanded universe who was Solo's teacher, so there's a good chance that at least some of this old character will rub off on the new one.
Digging into the history of Star Wars is tricky these days because technically most of it doesn't actually exist. However, we have seen some of the Star Wars Legends characters make the jump into official canon, and so it's possible that we could see something like that happen here. If Woody Harrelson is playing a character from the expanded universe there's really only one person it could be, an ex-bounty hunter named Garris Shrike.
In the late 1990s, a trilogy of books were published that dealt with Han Solo's entire life leading up to the events of Star Wars: A New Hope. The first of these, The Paradise Snare sees a young Han Solo as a member of the crew of a ship called the Trader's Luck, being captained by Garris Shrike. Shrike had built a small army of criminals, including many that he adopted as children and trained as pickpockets and thieves on the planet Corellia. Solo was an orphan that Shrike saw promise in, and so added him to the crew. Solo then learned how to be a thief and con man in this environment while he grew into a young man.
In the novel, Han Solo attempts to escape from Garris Shrike several times, eventually succeeding at about the age of 19. The young man goes on to live his own life, eventually becoming the man that movie fans know him to be. Solo and Shrike do cross paths one final time. Solo ends up with a bounty on his head, because of course he does, and Shrike attempts to collect it. It's the last time the two see each other. Or ever will.
Essentially, Garris Shrike is Fagin from Oliver Twist and Han Solo became the Artful Dodger. While the two characters had an antagonistic relationship for pretty much all of Solo's life, there's no question that Shrike would have to be called a mentor to him. If Woody Harrelson's character is coming out of the expanded universe, then this is who he's playing.
We're not necessarily postulating that Woody Harrelson will be playing this exact character in this exact story. In fact, it's highly unlikely this will be the case. There are a number of reasons that this doesn't quite work. Most importantly, the bulk of this story takes place when Han Solo is under 20 years old. Alden Ehrenreich is young, but he's not that young. While it's possible that some aspects of the Han Solo movie will deal with earlier points in time, we expect the vast majority to take place in a period more or less immediately preceding the first Star Wars movie. Still, they could very easily shift some of the events of a story like this until later in both of the characters' lives.
We've seen characters that were invented in the tie-in novels make the leap to the official canon. Grand Admiral Thrawn being the biggest example, so the idea of Garris Shrike making the same jump is far from a crazy idea. While Thrawn has only appeared in the animated series Rebels, the precedent still stands.
Even if Woody Harrelson isn't playing a character named Garris Shrike, he could still be playing essentially the same part. After all, Kylo Ren/Ben Solo as played by Adam Driver is a very similar character to that of Jacen Solo, the son that Han and Leia had in the novels who turns to the Dark Side. It's hard to believe that the character from the books didn't influence the one in the movies. Shrike and whatever character Harrelson plays could end up being so similar that they might as well be the same character, even if their names end up different.
However, this leads to one final question. If Woody Harrelson is playing Han Solo's mentor, is he also playing the film's villain? The idea of a mentor would imply that the two will be friends, but Han's mentor previously was somebody he ultimately wanted nothing to do with. If the movie and book characters end up with much of anything in common, it could very well be this.