Its source is an English expression meaning "A low-growing shrub with small purplish flowers."
Heather's spelling suggests the place where you might find the plant -- the rugged heath. However, the plant name evolved in Scotland while the name for the scrubby plain evolved in the south of Britain. In Scotland heather is found on the moor, not on the heath. On the heaths of England and Wales, you'll find a plant called ling, not heather.
Heather and heath did not find themselves in the same vocabulary set until the 18th century, when use of both words spread throughout the English-speaking world.
Note that the Latin name for heather is Erica, and it is possible some botanically-oriented parents have named their daughters with this in mind.