In the world of fandoms, there are always relationships. There are canon (ones that are actually in the movie, show, or book) ones and there are ones made up by the fans. The fandom pairing of these characters is called shipping. I am going to tell you some of the great non-canon ones for Game of Thrones.
1. Rhaegar and Lyanna
This is one that is hardly looked at. There is the bad side where people believe that Lyanna was kidnapped and taken advantage of by Rhaegar. I prefer the romantic theory. That she wasn't really in love with Robert and ran off with the Prince. Either way you look at it though, it ends tragically. Lyanna dies at the Tower of Joy and forces Eddard Stark to promise her something.
Some suspect that it was to take her son, Jon Snow, and raise him as his own.
2. Arya and Gendry
One of my personal favorite ships. They lived and travelled together for months. As time went by they became closer. At one point Gendry even said that she wouldn't be his family but his lady. To me they would be perfect together and balance each other out.
3. Brienne and Jamie
Lady Brienne of Tarth and Ser. Jaime Lannister definitely make sense together! Brienne helps Jaime by humbling him and Jaime lifts Brienne up and brings humor to her life. After the way that Jaime looked at the Sapphire Isle in a longing way makes many believe he does indeed have feelings for her. They may not both have beauty, but they connected on a deeper level, where Jaime and Cersei connect on a mostly physical (and sibling) level. I, for one, hope that Brienne and Jaime find true love within each other.
4. Bronn and Tyene
Ruthless cut throats and talented fighters just go together! Sure Bronn was almost killed by Tyene Sand, but that can be overlooked! Unless, of course, he gets super pissed about her mother killing Myrcella Baratheon! There is definitely some attraction between them and they're both full of themselves so something is bound to happen.
5. Daenerys and Jorah
I have been rooting for these two! I think Jorah should have told Daenerys the truth in the first place. Given how long it's been since he stopped spying there have been plenty of chances. Despite all of that he is truly and deeply in love with her. He is her protector and I think he deserves a second chance. He is from her homeland and Daenerys needs him. She possibly loves him whether she realizes it or not and that's why it hurts so much when she heard of his betrayal.