'Peckham' probably means the homestead by a hill; 'Rye' means a stream.
Peckham appears in the Domesday Book of 1086 as Pecheham, from Old English pēac, peak or hill, and hām, homestead. It has also been read as the village of the River Peck, but the stream's name is thought to come from the village's. 'Rye' comes from Old English rīth. In 1868 the vestry bought the Rye to keep it as common land.