(KJV) Samuel 16:3-4
3 And the king said, And where is thy master's son? And Ziba said unto the king, Behold, he abideth at Jerusalem: for he said, To day shall the house of Israel restore me the kingdom of my father. 4 Then said the king to Ziba, Behold, thine are all that pertained unto Mephibosheth. And Ziba said, I humbly beseech thee that I may find grace in thy sight, my lord, O king.
Then David hears the other side of the story
(KJV) 2 Samuel 19:24-29
24 And Mephibosheth the son of Saul came down to meet the king, and had neither dressed his feet, nor trimmed his beard, nor washed his clothes, from the day the king departed until the day he came again in peace. 25 And it came to pass, when he was come to Jerusalem to meet the king, that the king said unto him, Wherefore wentest not thou with me, Mephibosheth? 26 And he answered, My lord, O king, my servant deceived me: for thy servant said, I will saddle me an ass, that I may ride thereon, and go to the king; because thy servant is lame. 27 And he hath slandered thy servant unto my lord the king; but my lord the king is as an angel of God: do therefore what is good in thine eyes. 28 For all of my father's house were but dead men before my lord the king: yet didst thou set thy servant among them that did eat at thine own table. What right therefore have I yet to cry any more unto the king? 29 And the king said unto him, Why speakest thou any more of thy matters? I have said, Thou and Ziba divide the land.
After hearing both sides of the story, still David allows Ziba to inherit some of Saul's Possessions.
Could it be because Ziba had met the king with some provisions (2 Samuel 16:1-2).