‘Sydenham’ was once Chipeham, probably the farmstead of a man called Cippa.
It was recorded as Chipeham in 1206, and the name changed in stages over several centuries; for a long time the place was called Sippenham. The last step, when the p became a d, came in the late 17th century according to Hidden London, while another local history says the name was Sippenham until the 18th century.