The following is from three sites about names and meanings:

• From a Scottish surname, which was derived from a Scottish place name, probably derived from Gaelic leas celyn meaning “garden of holly”. It has been used as a given name since the 19th century. In America it was more common as a feminine name after the 1940s.
• It is of Scottish and Gaelic origin, and the meaning of Leslie is “holly garden“. Also possibly “the gray castle”. Place name and name of a prominent Scottish clan, first used as a given name in the 18th century.
• Transferred use of the Scottish surname taken from Lesslyn, a place-name in Aberdeenshire. The name might be derived from the Gaelic elements lios (enclosure, garden, fort) and chuilinn (a holly tree) or liath (gray): hence, “garden of hollies” or “the gray fort.” The name, borne by a well-known Scottish clan, was not in common use as a personal name until late in the 19th century.

Basically, it comes done to two meanings: Holly Garden and/or Gray Fort.

Holly garden is interesting because holly is used to symbolize truth in heraldry. So truth becomes part of the meaning. The idea of a gray fort is connected personally to my habit of standing back and observing, like a fort on a hill. I am also very closed off at times, like a proper fort.

Fun fact: In the Harry Potter novels, holly is used as the wood in the titular character’s wand.