Alyssa came into use most likely in the 1800s along with the trend towards flower names. The trend of using feminine names with fancy suffixes like -inda and -issa (for example, Clare -> Clarissa, Clarinda, Lucy -> Lucinda etc.) was also in place.

Alyssum is a flower once thought to cure rabies, hence the name, which is Greek for 'not insane.' Alyssa, a variant on Alice, was most likely inspired by this flower for the spelling.
See Also: Allyssa, Elyssa

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