The Edward o'malley valve co, of Chicago made the 'Nu-Seat' Valve insert for faucet repair, it was a brass insert that screwed into the faucet in place of the original cast in seat. The tool(s) bored out the old seat, and threaded the edges , and the brass 'nu-seat' insert was screwed into the faucet. They made various levels of kits including a metal boxed app in one kit, down to simple t-handle home owner kits. The simple kits are advertised in Pop Sci in the late 40's-50's; The nu-seat insert is listed as 'new' in the 1948 edition of Milk Plant Monthly.
I am assuming yours, with a wood box , is early , as the references I see list a metal box later...
Did not find anything matching exactly that kit tho...
The company may know more...they are still around, not bad for a 1910 company ;)
http://www.omalley.com/about/