We read a few times about the lameness of Mephiboshet, who was the last remaining progeny of Saul. He was the son of Jonathan, David's closest friend, and David sought him out in order to provide favor to him. The type of favor that he provided was one where he actively sought out any of the previous king's family members and wanted to give them what they had in the past.
As you can imagine this is incredibly unusual. The normal course of action is to find any of the previous ruler's family and eliminate them so that they would not be a threat to the current ruler. But David, in spite of being a serial sinner, has a hear for what God wants. He always was wanting to be sure that Saul was not harmed, at least not by him. So David spends a lot of time and effort to ensure that Jonathan's son gets favoritism, even to the point where he eats every meal with him.