The origin of the name Alyssa can be traced back to two different sources, neither of which are Italian.
The first is as a derivative of the name Alice, which has Germanic roots and means 'of noble kind' or 'of noble birth'. In Italian, the name Alice is spelled in exactly the same way as in English but is pronounced [ah lee chey]. The word Alice is also Italian for anchovy, although acciuga is more often used when referring to anchovy as a food product.
The second meaning for the name Alyssa derives from the genus of the garden plant Alyssum of the family Cruciferae. In this context, Alyssum derives from the Greek word lyssa meaning madness, rage or fury, and could be linked to the supposed mythological property of the flower: Its reputed ability to cure madness.