He chooses Percy, a member of a solid wizarding family, in order to stay aware of what was going on in the magical world and perhaps hear word of his former master, Voldemort. Not to mention the fact that there was plenty of good food and places to hide at the Burrow.
How Pettergrew managed to insinuate himself into the Weasley family as a pet is unknown. We do know that Pettigrew faked his own death on about November 1, 1981, and escaped the scene in rat form. He remained hidden as a pet until he was exposed in 1994.