There are a lot of misconceptions about sign language in general. Let's clear some of the big ones up before we jump in.
Now, about the history of American Sign Language.
The first permanent school for deaf children in the United States was opened in 1817 in Connecticut. Its first teacher was Laurent Clerc, a deaf man from France, whose native language was French Sign Language (known as LSF). The school started with seven students, and Clerc taught in a combination of gesture and LSF, along with written English.^1
We might all be using LSF today, except that when Clerc came to Connecticut, there were already multiple regional sign languages in use. The best known of these is Martha's Vineyard Sign Language, which was used in the 17th-19th centuries while there was a high population of hereditary deafness on the island.^2 Because transportation was fairly limited during the 17th and 18th centuries, most people on Martha's Vineyard stayed there for most of their lives; the high rates of intermarriage perpetuated the high deaf population for decades. As transportation improved, though, deaf students from MV began attending the school in Connecticut, bringing their own sign language and intermingling it with the LSF already in use there. (For more on Martha's Vineyard, see Everyone Here Spoke Sign Language by Nora Groce.)
Additional linguistic contributions came from Maine, as Sandy River Valley Sign Language was used there and brought to Connecticut, and from Henniker Sign Language, used in New Hampshire. Even students who had not been in a signing community previously contributed to the linguistic makeup of ASL, as they used "home signs" at school. No one language could be said to be dominant in these early years of the language; students used whatever signs they knew, and as long as they could make themselves understood, they retained those signs. If another student had a more clear sign to represent a concept, the student would adopt that sign. (See Susan Goldin-Meadow, The Resilience of Language.)
Eventually, all this language sharing grew into American Sign Language. Although Clerc originally used LSF, along with his students, he transitioned to ASL as it developed. Some of his early students were adults themselves, and they took the new language with them as they became teachers at schools across the country. Clerc also traveled across the United States, and his teachings became the foundation for over 20 schools during the 19th century.^3
Today, ASL and LSF have only about 50% lexical similarity. As ASL has evolved, so has LSF. Bonnal-Vergès (2006) notes that LSF today is more complex than it was around the time of Clerc. Although I have seen videos of ASL from the 1910s and I can understand them just fine, I have not seen historical videos of LSF to determine when the divergence became significant.
A final note: while I have heard that Native American signed languages may also have had influence on the early years of ASL, I have no citation for this. It does not seem to be referenced in Jeffrey Davis's Hand Talk: Sign Language Among American Indian Nations which is one of the chief references on Native American signed languages. I have not read enough of Melanie McKay-Cody's work to know if it is mentioned there, but I cannot speak authoritatively on the subject as I am not well versed enough in Native American signed languages.