How come Littlefinger spent his childhood at Riverrun? I understand that the children of Lords are often sent as wards to other houses, for political reasons (eg. Robert Baratheon and Eddard Stark at the Eyrie). But Baelish is a much lesser house than Tully. Who sent Petyr to Riverrun, why did the Tullys agree to take him?
During a signing tour, George R.R. Martin responded to this question by stating that Petyr's father befriended Lord Hoster Tully during the War of the Ninepenny Kings.
This resulted in Petyr's father being able to arrange for Petyr to be fostered at Riverrun, where presumably he hoped that Petyr would have more options and opportunities than if he were raised on the rather desolate rocks they called home.
Fostering another noble family's children is a Westerosi custom especially prevalent among the stronger houses. There are several reasons why a house would send a child to be fostered elsewhere:
To strengthen the bonds between the two houses. Having the children of two houses grow up, play and learn together increases the chances that the two houses will remain allies when the children grow up and become lords themselves. Both Ned Stark and Robert Baratheon were fostered at the Eyrie under the tutelage of Jon Arryn. A friendship that turned into an alliance between the Baratheons, Starks and Arryns when Robert took up arms against the Targaryen kings.
To insure a defeated house's good conduct, through keeping the foster child as a hostage in all but name. Theon Greyjoy was fostered at Winterfell to insure his father keeps the peace after his failed rebellion.
As an honor bestowed upon a lesser house, which is the case for Littlefinger. Peter's father became a good friend to Hoster Tully (his liege lord), and as a token of friendship Hoster agreed to foster Peter at Riverrun. There he would get better education and training than he would in his father's poor holding. It would also serve to get him better connections by meeting and befriending other noblemen frequenting the castle, which could only help further his future.