No one person invented sign language - it evolved world-wide in a natural fashion, much the way any language evolved. We can name a few people as the innovators of specific signing manuals. Each language English, French, German etc developed their own respective sign languages at different times. American sign language (ASL) is closely related to French sign language.
In 1620, the first book on sign language that contained the manual alphabet was published by Juan Pablo de Bonet.
In 1755, Abbe Charles Michel de LEpee of Paris founded the first free school for deaf people, he used a system of gestures, hand signs, and fingerspelling.
In 1778, Samuel Heinicke of Leipzig, Germany founded a public school for deaf people, where he taught speech and speechreading.
In 1817, Laurent Clerc and Thomas Hopkins Gallaudet founded America's first school for deaf people, in Hartford, Connecticut.
In 1864, Gallaudet College, in Washington, D.C was founded, the the only liberal arts college for deaf people in the world.