Game of Thrones has Tyrion visiting the Wall, simply as he wanted to see it.
Watching this, I had the feeling that, in the novel, there was another, less arbitrary, reason, but I can't recall it.
Am I remembering correctly, or was Tyrion simply being a curious tourist?
I have the book in front of me. There was no reason mentioned other than Tyrion wanting to satisfy his curiosity. Or as he put it:
"... No, I just want to stand on top of the Wall and piss off the edge of the world."
Which, amusingly, he does in the HBO series :)
This appears to be answered further in A Dance with Dragons.
Tyrion was given a book "Wonders and Wonders Made by Man" that describes the 7 wonders of the world made by the gods, and the 9 wonders made by man. He read these books 'until they fell to pieces' and had 'committed all sixteen ... to memory as a boy', and 'cherished a dream that one day he would travel the world and see [them] for himself'. The Wall is not directly referenced as one of these wonders, but it is a fairly good contender for one of them. If the stone roads of Valryia are considered a wonder, it is a fairly good chance that the Wall is.
I think that George Martin "sent" Tyrion to the wall because in later stages of the story, the wall, and manning it, becomes a major storyline, and in a way, Tyrion will become the Wall's ambassador to King's Landing and one who can understand what the wall really is and why it is needed.
To the best of my knowledge, he was playing the part of a tourist.
It's not hard to see why Tyrion would want to see The Wall. It would be the greatest feat of engineering in Westeros. With Tyrion being a learned man, it's no surprise that he jumped at the opportunity.
Tyrion Lannister reads all that he can get his hands on in Game of Thrones. He is usually buried in a book about history, dragons, cities, whatever he is doing, whatever task he has been given, or just for knowledge itself. He says he has been fascinated with seeing the 9 Wonders Made by Man since he was a young boy. The wonders include the Wall, The Titan of Braavos, The Great Pyramid at Ghis to name a few.
Tyrion meets Jon Snow at Winterfell in S1 of Game of Thrones, "Winter is Coming." This happens in "A Game of Thrones," the first book in "A Song of Ice and Fire." His interest in Jon Snow would have begun here.
King Robert Baratheon has come to Winterfell with his wife, Queen Cercei, their children, her brothers, Ser Jaime and Tyrion Lannister after the death of the Hand of the King, Jon Arryn.
The HBO episode, "Winter Is Coming," is a moving first episode that sets up the Houses Stark, Baratheon and Lannister. It introduces us to King Robert's world, The Stark's beautiful, loving family who are about to be plunged into another world entirely. It also establishes the deep friendship between King Robert and Ned Stark, the love Robert had and still has for Lyanna Stark, Ned's sister. Telling Ned that she should be buried where he can be near her, in the Sun, not down in the dark crypt in which she is entombed. He speaks of his hatred for Prince Rhaegar Targaryan and his wish to kill the surviving Targaryans. How he kills Rhaegar every night in his dreams. The King visits Winterfell to ask Ned to be The Hand of the King in Jon Arryn's place. There are touching exchanges through this episode that show just how the Lannisters will affect the future lives of the noble Starks. The extreme difference between the two families is extremely apparent. Tyrion, being the watcher & observer that he is, notices how uncomfortable (and drunk) Jon Snow has become throughout dinner, as has Tyrion himself. They then come across one another outside the hall. Jon has just run out of the dinner, embarassed and in pain about a comment made regarding his status as a bastard. He was made to sit at a different table from the rest of his family at Catelyn Stark's order. She would not have a bastard at her table in front of the King.
The pain Tyrion feels being a dwarf and an outcast, and the pain Jon Snow feels at being a bastard, as well as the way he is treated by the only mother he has ever known do seem to coincide nicely. It doesn't seem a direct connection, but one in which the two outcasts find themselves thrown together outside of a dinner where all are related or are of royal blood or marriage. Even though Tyrion is a Lannister and a Lord, he still never feels accepted or loved by his family, he states this in his words and his behavior. Jon has had a wonderful childhood as a Stark, but knows he can never be anything but a bastard. Perhaps out of a feeling of kinship; always opinionated, Tyrion starts talking to Jon Snow. This exchange happens in the book, "A Game of Thrones," as well. Tyrion tells Jon this:
"Let me give you some advice, bastard, Never forget what you are, for the rest of the world will not. Wear it like armor, and it can never be used to hurt you."
This is from, "A Game of Thrones," George R.R. Martin's writing is slightly
different, but the difference is an important one:
Never forget who you are, for surely the world will not. Make it your strength, then it can never be your weakness. Armor yourself in it, and then it will never be used to hurt you."
There is quite a different feeling to the second quote. The writing of George R. R. Martin, is a bit, softer.
Tyrion does seem to feel a kinship with Jon Snow, he feels something close enough to compassion to bother to say this to Jon at all. There is something to see in Jon Snow. He has a quiet power. He is bravely going to the Wall because he knows that a bastard cannot get very far in Westeros. That his title will define him enough to hinder him from doing much unless he goes to the Wall. There, it doesn't matter. He's just another brother of the Night's Watch. That kind of bravery from all of the Night's Watch might be interesting and intriguing to Tyrion, he may want to have the experience of meeting men who would make a vow as serious as the one the brotherhood of the Night's Watch make.
There are times that Tyrion seems nonchalant, but Tyrion is not nonchalant. So much of what Tyrion does and says is to get a reaction or for shock value. As with his inappropriate drunken comments and constant whoring. It is all a defense mechanism. He is extremely witty and usually drunk. So, Tyrion could have simply gone to the Wall for the very reason he said he wanted to go;
" To piss off the edge of the world."
But Tyrion is smarter than just that. It's not clear what his reasoning was, but there is much to be curious about beyond the Wall, and at the Wall itself. Tyrion has a great deal of curiosity. He knows magic exists in his world, he knows that the Wall is a mysterious structure. That it was built somehow, there are tales no one knows of which are true.
The reality of why Tyrion went to the Wall? To see it, to see what the Night's Watch were like, to talk to them, to hear stories, to know what is out there. To prepare himself for what they will need to defend themselves against when Winter comes. The pair may meet again in fan speculation, but, we will just have to wait for "Winds of Winter" and Season 6 to see if they do.
Remember, in Westeros there is usually more to a thing than it seems.
Disclaimer: I have now seen just the first three episodes of the TV series and have not read any of the books. From what I have seen, I would hazard that:
Tyrion is taking an interest in the career of Jon Snow. He has been goading him about being a bastard, and obviously relishes it when Jon says that he is the only one who told him the truth about the wall. Tyrion makes sure that he shows up to save Jon from being beaten up, but also to make him question everything he thought he believed. Tyrion is courting him as an ally against the Starks by making him feel betrayed by them.
Also, he appears to have used his position to have the Lord Commander intercept Snow's raven. Obviously he is watching for news of whether Bran has woken, and what he might say. Given that he was strongly implicated with the dagger, it may be that he thought it was safer to intercept that news on the Wall. Certainly safer than in Winterfell, and possibly even safer then he would have been in King's Landing, where Ned Stark is the new Hand.
Edit: I am now partway through the second season. In the next episode after the Wall, when Tyrion returns from the Wall and passes through Winterfell, he does the same thing with Theon - probing him for his loyalties to the Starks.
Tyrion is a complex character, and his loyalties are not entirely clear. He says "I don't know war, but I know people." I think the best answer is that he goes to the Wall to gain knowledge about the kind of people who are there, including or especially Jon Snow. In season 1, Episode 103 (the third "production notes" bonus material), the actor who plays Tyrion says "Tyrion relates to Jon Snow because John is the black sheep of his family as well. Tyrion is treated like a bastard, so they have that in common."
Edit: I am now partway through the fifth book, A Dance with Dragons. There, Tyrion explicitly recalls as a youth having been fascinated with two books given to him by an uncle. The books by Longstrider were accounts of the natural and man-made wonders of the world. He says that he had dreamed of being able to visit all of them, but that his father deliberately crushed those dreams.
The Wall has been twice identified by GRRM as being one of the man-made wonders. (See http://awoiaf.westeros.org/index.php/Wonders_Made_by_Man)
Thus it seems likely though not explicit that Tyrion is taking the opportunity to fulfill one small part of his childhood dream of visiting the wonders of the world.
