Evidence has been produced that supports the theory that our name was originally spelled "Siggeswick" or "Seggeswick." The village of Sedgwick in Westmorland used to be called Seggeswick. A map from the mid 1600's shows this spelling.
Article: Derivation of the name Sedgwick

The image below is a portion of a map of "The Countie Westmoreland and Kendale the Cheif Towne" [sic]. The map was drawn by John Speede between 1610 and 1623. The image is from a reproduction purchased from The Old Map Company of Great Britain and used by permission.

Compare to this modern map.

SEDGWICK, a township in the parish of HEVERSHAM, KENDAL ward, county of WESTMORLAND, 4 miles (S.) from Kendal, containing 184 inhabitants. There is a place of worship for Independents. A mill for the manufacture of gunpowder was established here about 1770. The river Kent and the Lancaster canal pass through the township. Here is a free school with a trifling endowment.

-A Topographical Dictionary of England, Samuel Lewis, S. Lewis and Company, London, 1831, Volume 4 page 33, available at Ancestry.com