The story about Henry V giving the "Tripp" name to Richard Howard appears on p. 1433 in Vol II of Burke's 1847 "A Genealogical and Heraldic Dictionary of the Gentry of Great Britain and Ireland." Burke cites older documents including the autobiography of James II. Various versions of this book can be found on the Internet Archives and Google.
Henry V was born 300 years after the Domesday Book was completed in 1086 leaving 15 generations of Howard/Tripps in between to account for. I doubt if my Tripp ancestors even had a surname 800 years ago.