San Jose was founded by Father-President Fermin Lasuen, who succeeded Junipero Serra as head of the mission chain. He wanted to fill in the gaps between the established missions and planned to establish five new missions.
An exploratory party of six or seven men accompanied by a padre from San Antonio de Padua was assigned to find a good site for one of these missions in the area below San Francisco. The party traveled northeast from the existing mission of Santa Clara. The picked out the site pretty quickly, but actual approval and funding took a couple more years.
Fr. Lasuen was a hands-on leader and he personally led the group that founded the mission, setting out from Santa Clara on June 9, 1797. Two days later they erected a cross on the site. Fr. Lasuen decided to name this mission in honor of the foster-father of Christ, St. Joseph.