Answer:  

It seems that your information is substantially correct. The name Renton hails from the parish of Coldingham in Berwickshire (Granthouse is just 8 miles away). Blacks's Surnames of Scotland states that  "The Rentons of Renton were hereditary foresters of Coldingham in the reign of William the Lion, and Richardus forestarius appears in record as Ricardus de Reningtona." That branch of the family terminated in the direct line in a daughter during the 15th century.

 

 

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