Of German roots, Evelyn is an English variant of Aveline. Aveline is itself related to Ava, Medieval short form of Germanic names beginning with the element "avi", of unknown meaning, possibly "desired".
It can also be considered a cognate of Eve and Lyn - then of Hebrew origin. There is a possible relation with the Gaelic 'eiblin' ('pleasant') as well.
The medieval name seems to have been used only by women, but it gave rise to a surname that was turned into a masculine given name as early as the 17th century. Evelyn is today used primarily by women, but it was common enough as a boys' name just a generation ago.
An example of it as a surname is the gardener and diarist John Evelyn. Novelist Evelyn Waugh is a famous male bearer, and Pre-Raphaelite painter Evelyn De Morgan is a female bearer.
In the 20th century, Evelyn was most popular in the 1910s and 20s. Recently it has been resurging.