Game of thrones theories and predictions :

1. Tyrion Lannister is actually Tyrion Targaryen:

This may not be evidence based but it atleast holds a bit of merit. Tyrion has always been an outcast from his own Lannister family. Tywin once commented that Tyrion is “No son of mine”( it could also be persiflage). Also tyrian has the same color of hair as many of the Targaryen. Also he and Jon Snow had a sense of kinship in the first season.

2. Jaime will kill Cersie:

According to the prophecy of the foreteller, Cersie will lose all her children(and she actually does) and she will be killed by her younger brother. While we think of Tyrion to do this sororicide, we should not forget that Jaime is also younger to Cersie. Jaime, who is the ‘ king’s slayer ‘ might also slay the queen.

3. The fourth dragon egg:

There is another dragon egg and it is believed to be deep under the ocean.

4. Syrio Forel is alive as Jaqen Hghar:

 Although  Syrio was seen last fighting with the lanister men with a broken wooden sword, he was not shown dead.

Syrio and Jaqen both said that they are from Bravos. Also as Jaqen can change faces, it might be possible that he is actually Syrio changing faces and protecting and training Arya.

The person behind the girl in Bravos looks like Syrio.

Also both Syrio and Jaqen called Arya a boy.

5. Reunion of Arya and her wolf:

Arya separated from her wolf when it attacked the prince of kings landing(fearing it might be killed). There would be great eagerness to see the direwolf save Arya when she is in danger.

6. Jon Snow going North to catch a living white walker:

To prove that the white walkers exist , Jon Snow might go north to catch one. The aim of this might be to reunite the kingdoms against their strongest enemies.

7. Making of swords from dragon stone: 

The white walkers can only be killed with dragon stone. So some one is definitely going to make swords from dragon stones.