Some people think that Juan Pablo de Bonet invented sign language. In the year 1620, Bonet wrote a book that contained the first known manual alphabet system. The handshapes in this system represented different speech sounds. This is a great accomplishment, but realize that he created the first known manual alphabet system, not the first manual alphabet.
Still looking into who invented sign language, some people think that the deaf people living on Martha's Vineyard invented sign language. Martha's Vineyard Sign Language (MVSL) is an early sign language that was used on the island of Martha's Vineyard that is off the coast of Massachusetts. Hereditary deafness was very common on this island starting in the 17th century. These deaf people were not the first to use sign language (Bonet came before them), but they did have an impact on the schools for the deaf that were soon to come.
Abbe Charles Michel de L'Epee is a common answer people give when they are asked, "Who invented sign language?" L'Epee established the first free public school for the deaf in Paris in 1771. L'Epee educated the deaf using a standard sign language that he created. So, people obviously thought he invented sign language since this was the first sign language that was used widely.
However, they fail to look at the fact that children came from all over the country to go to L'Epee's school (some even from Martha's Vineyard). The deaf children had been signing at home (because of lack of other communication), and brought these signs with them. L'Epee learned all of these different home-made signs from the children and created his standard sign language based off of them. Just like Bonet, L'Epee created a system, but not the first language. L'Epee's standard sign language eventually became French Sign Language and was widely used in Europe.
The first American school for the deaf was established in 1817 by Laurent Clerc and Thomas Hopkins Gallaudet. They are often credited as the inventors of American Sign Language. This is actually partially true. Laurent Clerc was from Europe and taught French Sign Language. Thomas Hopkins Gallaudet brought Clerc back to America to start the first American school for the deaf. Like Abbe Charles Michel de L'Epee's school, children from all over the country traveled to attend this school, bringing their home-signs with them. These home-signs, combined with French Sign Language, became American Sign Language.
Like any deaf person would, the children who attended the schools for the deaf created a language to communicate. Spoken language is not the natural language of the deaf. They naturally defaulted to their native language--sign language. Therefore, no one person invented sign language.