Glynnis
One of my very good friends. Her nickname is Glyn or Glynnie. I feel it is a very Irish or Scottish name. Glynnis Clare is a very cute double name as well.
Glynnis is a variant of Glynis, which probably is a variant of the Welsh name Glenys, changed to reflect the Welsh word glyn, meaning "valley." Glenys is derived from glān meaning "holy; clean; pure."