In the original series episode Journey to Babel, (and only what is seen on screen is considered canon) Spock’s mother tells Kirk she can pronounce Sarek’s family name after a fashion. Since it’s a family name, that is what would be Spock’s other name. But it’s not mentioned in the series.
Although Trek novels are not canon, in the original series novel Ishmael, it is revealed that Spock (and Sarek’s) other name is S’chn T’gai if I recall correctly. This is seen near the end of the book where Kirk and McCoy are doing a little research on Spock’s family tree.
