The sauce shares its name with a Mexican state, and nobody is sure what that name means.
Edmund McIlhenny made the first batches on Avery Island, Louisiana, after the American Civil War. Mental Floss says he planted his first pepper crop there in 1868 and first packaged the sauce in a used cologne bottle, later ordering new cologne bottles for sale. Tasting Table says he mixed Capsicum frutescens peppers with salt from the island, fermented the mash and added vinegar. It reports that he received a patent in 1870.
The origin story is itself disputed. Wikipedia says the company credits McIlhenny with the first sauce in 1868. Other accounts say a New Orleans planter, Maunsel White, was growing tabasco chilies by 1849 and gave McIlhenny pods and a recipe before White died in 1862. Wikipedia adds that McIlhenny only won the right to use "tabasco" as a proprietary name in 1948, after decades of litigation.
Tabasco is the name of a state in Mexico, and that is where the word comes from. Etymonline says the place name comes from an unidentified indigenous Mexican language and its etymology is unknown.
Several meanings have been offered. The company says the name may mean "place where the soil is humid" or "place of the coral or oyster shell." Mental Floss adds that other accounts link it to a river, a Mayan phrase or a Nahuatl term for flooded land. None of these has been settled.
Why did McIlhenny pick the name? Tasting Table suggests the simplest answer. The pepper he used is commonly called the Tabasco pepper, because it grows abundantly in the Mexican state, so the sauce took its name from the pepper. Etymonline says the name may come from the state because European travelers first met the pepper sauce there. Neither source records a statement from McIlhenny himself.
Etymonline dates the sauce name to 1876 in English, when it was a proprietary name. A trademark registered by Edward Avery McIlhenny claims use from around 1870.
A popular claim is that the seeds came from Mexico. None of the sources we read says so, and Wikipedia says only that the peppers were first grown on Avery Island. What the sources do support is the link between the pepper, the state and the sauce.