put one’s name down for something—
1. apply for a position, etc.:
- He was called up having put his name down for the RAF and after initial training he was sent overseas.
2. promise to subscribe for a certain sum:
- One man after another said that he would be willing to put his name down for five or six pounds.
put one’s name to something—sign a declaration, etc.:
- The former assistant Chief of Defense Staff put his name to the appeal, alongside retired general Sir Hugh Beach.
See also: give a name to something / give one’s name to something.