It is a locational surname from Pagham, Sussex, or Pakenham, Suffolk.Antonia Pakenham, maiden name and early pen-name of author Antonia Fraser
Edward Pakenham (1778–1815), British general
Sir Francis Pakenham (diplomat) (1832–1905), British envoy to Chile, Argentina and Sweden
Text adapted from Wikipedia, CC BY-SA 4.0. Photograph: Philip Mallis from Melbourne (see Commons), Wikimedia Commons.