No one town can claim mariachi

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Today the state of Jalisco claims the mariachi. There are actually a few places in Jalisco that claim that the mariachi tradition originated in that particular town. There's a song called "El Mariachi de Cocula"- the mariachi from Cocula. Cocula is a town in the state of Jalisco, not far from Guadalajara.
In the Coca language there actually was a town. It was named "Cocullan" and then at some other point it's "Cocolan" and then "Coculan" and "Cocula." Today there's a small museum there - a small mariachi museum there. But there's also a small mariachi museum in Tecalitlán and other places in that region.
The mariachi was not born in any one particular town. It is of a region, it originated in a region that included the states of Jalisco, Nayarit and Michoacán. There were a number of different groups that called themselves mariachis, using a variety of instruments during the 1800s and into the early 1900s that are no longer part of the mariachi tradition today.
Incidentally, mariachi was also the name of a town. A Spanish census of some time back showed the town of Mariachi with a population of 90. No one knows where that town is today but it certainly existed in the ledgers.