Most clothing brands use a surname. Levi's uses a first name.
The man behind it was born in 1829 in Buttenheim, Bavaria. Several biographies say his birth name was Loeb Strauss and that he took the name Levi after emigrating. He emigrated to the United States and joined his brothers' dry-goods business in New York. In 1853, he moved to San Francisco during the Gold Rush and set up a wholesale firm under his own name. An 1858 city directory lists the business as "Strauss, Levi."
The firm later became Levi Strauss & Co. and was incorporated in 1890 with his nephews. It did not start out making jeans. It was a wholesale dry-goods business.
The product came through a customer. Jacob Davis, a tailor in Reno, Nevada, bought fabric from the firm and began reinforcing work pants with metal rivets at stress points. In 1872 he wrote to Strauss. Davis could not afford the patent paperwork, so Strauss paid for it. They were granted a patent on May 20, 1873, for riveted trousers.
The nickname form is a later habit. Etymonline records Levi's in 1926 in American English, as the name for pants made by the San Francisco firm. The pants were sold as "waist overalls," and the lot number 501 was assigned in 1890.
Nobody knows why the number 501 was picked. The company's records were lost in the 1906 San Francisco earthquake.
The fabric also has a place-name story. Etymonline says denim comes from French serge de Nîmes, "serge from Nîmes," and that denims as trousers are recorded from 1868.
I could not find a source on exactly who first shortened the company name to Levi's. The earliest dictionary record is from 1926; this page found no earlier record.