In S04E05 of Game of Thrones, Lady Sansa Stark and Lord Petyr Baelish arrive in The Eyrie. They are greeted by Lysa and Robin Arryn. Robin is surprised (strange, considering he was there only months earlier). He calls him "Uncle Petyr". I wondered for a long time why this was. Was it a term of affection? Did Lysa tell Robin to call him that because he grew up with the Tully's? Or does he simply think that he is a relative because he acts like one?
This is speculation, but I would assume that it is simply because Lysa told him that is who he was, in the broadest sense of things.
As a child in English culture, it's not uncommon to for parents to introduce their friends to their children as uncles or aunts. It is much less complicated for the child to know instantly that this is a person that they can trust as they are within the family, rather than try to explain to them "this is one of my friends".
Once the child is grown past being a toddler they will understand that the person isn't related to them and often will stop referring to them as uncle or aunt, though sometimes it will stick as a term of affection for the person.
This leaves us with three options for why Robin calls Petyr uncle:
Robin has grown up referring to Petyr as an uncle and truly feels as if he is as close as a member of his family, so carries on using it as a term of endearment.
Lysa encourages him to use it to make Petyr feel as if he is part of the family, possibly so he'll visit more often if he feels closer to the child and incorporated into the family.
Robin isn't quite as advanced as he should be (for example he still breastfeeds and doesn't quite understand the concept and finality of death. He thinks the Moon Door is for fun, rather than for execution). Therefore he may actually not understand yet that Petyr isn't his real uncle, and simply still calls him that because he believes he is related to him.
"Uncle" here is being used as term of very close familial ties, not because Petyr is actually family, but because Lysa wants him to be.
Robin's mother, Lysa Arryn, is in love with (or, perhaps more accurately, infatuated with) Petyr Baelish. She would have preferred to marry him, but was basically sold off to Jon Arryn as part of his support for Robert in the rebellion. With her husband dead, she continues to hold onto the fantasy of marrying Petyr and living out the rest of her life with him.
Petyr has been taking advantage of that to ingrain himself into the noble court in the Eyrie. He plays up the idea that he and Lysa could have been a happily married couple, with a son of their own, because he knows it reinforces the fantasy she has about her future with him.
Lysa wants her son to have a close relationship with Petyr, so she insists that he is "Uncle Petyr". This plays right into Petyr's plan, since Robin is the heir to the Eyrie, Petyr now has a pretty strong influence over the boy. The term here is being used as a kind of "generic male adult relative"; obviously he can't call Petyr "Father", so "Uncle" is the next best thing.
It may be a subtle clue that Petyr still loves Catelyn, not Lysa. If Petyr and Catelyn were married, Petyr would be Robin's uncle. It could be hinting that because of Petyr's continued feelings for her, in his mind, he's Robin's uncle, not his father. I'm not sure how Robin would know that, but it's an alternative theory.