The most plausible explanation of the name is that Henry Wood and his family moved there from Bury, Lancashire, England, in 1683. As was custom of many early settlers who were first arrivals at a spot, Wood combined his name with that of his former home, calling the place Woodbury." (McMahon, 1973)
1808 map "Plan of the City of Philadelphia and Environs" surveyed and published by John Hills also showing Woodberry spelling.
McMahon, W. (1973). South Jersey towns. New Brunswick, NJ: Rutgers University.