I watch the Game of Thrones TV series and I play the strategy game Game of Thrones: Ascent, but I have not read any of the books other than the first chapter of the first book. In Game of Thrones: Ascent, there is a quest line to reincarnate, starting the game all over, allowing you to keep only certain things. Does this stem from a concept of reincarnation in the books or is it just a feature of the game?
We see the spirit of the skinchanger Orell live on inside the body of his eagle.
In the prologue to ADWD, we see the skinchanger Varamyr Sixskins transfer his spirit into his wolf companion.
So, can you die and come back in ASOIAF? Yes, you can. Can you be reincarnated as someone or something else, like Beric being reborn as a butterfly? No. Can your spirit be transferred to other bodies? Yes.
The red priests of R'hllor have the ability to
bring people back from the dead, such as Beric Dondarrion and Catelyn Stark.
Whether or not that counts as re-incarnation is highly debatable.
Re-incarnation? Not so much in the exact sense of the word; therefore it seems that that is a feature of the game only. (Barring the ability for certain characters to be re-animated using magic by the Red Priests).
However, consider that there seem to be many parallels between historical figures and current storylines of certain characters.
The Prince That Was Promised
Azor Ahai seems to be the closest contender for this title; Azor Ahai was a hero who lived in the Age of Heroes and was believed to have lived during the Long Night and was responsible for defeating the Others.
The Prince That Was Promised (TPTWP) is said to be the return of this hero. Rhaegar Targaryen believed that this hero was to be born of his line (he believed it to be his son Aegon VI), whereas many believe this true return of this hero to be
TPTWP was mentioned in The House of the Undying during Daenerys' visions, where she sees her deceased older brother Rhaegar conversing with a woman (believed to be his wife, Elia Martell), and speaking of their newborn son (believed to be Aegon VI) saying:
He is the prince that was promised, and his is the song of ice and fire.
-A Song of Ice and Fire: Book Two - A Clash of Kings, Chapter 48 (Daenerys IV).
The Night's King
This one is a slightly less definitive one. No one knows the true purpose of the original Night's King; but many are likening Stannis Baratheon to the original Night's King for his "unholy" union with The Red Woman (Melisandre).
It is unclear whether or not The Night's King was the same character as The Great Other (TGO), but there definitely seems to be a growing climax towards a clash between TPTWP and TGO. Whether or not TGO will become manifest, and if so, whether or not it will be the same ageless character, or a re-incarnation is yet to be discovered.
Yes, its theme found its way onto the show in Season 5 Episode 7. Maester Aemon of the knights watch says in a different voice "Egg! I dreamed that I was old" on his death bed. It is theorized that the masester had just opened his eyes as the newly reincarnated boy without fully dying yet. This answer is my favorite as it may not be cannon as it's drawn from the prospective of the TV show and without the books.