I like
Lynette very much - always have, though I too don't like
Lynn.
Lynette has history behind it since it's the name the poet laureate Lord
Tennyson used as a form of the Welsh
Eluned (who appeared in Arthurian legend) in his poem "Idylls of the
King" based on the
King Arthur and his knights saga. I think
Eluned appeared as a version of
Lynette (
Linette or something similar) in medieval French versions. It is not unlike the modern Welsh name
Llinos ("linnet").
It's a name that's not heard too often and I think it deserves more attention.