M. Barr and J. Beck, Acyclic models and triples, Proc. Conference Categorical Algebra (La Jolla, Calif., 1965) Springer, New York, 1966, pp. 336-343.
The "triple" is an older name for a monad.
Also, Jonathan Mock Beck is the same as Jon Beck, as you can see from these lecture notes and this Wikipedia page.
Follow-up: In Beck's thesis the monadicity theorem is discussed on page 8 with reference to an unpublished note: J. Beck, The tripleableness theorems. To appear. The editor's note adds that "To our knowledge, this has not appeared. Beck’s tripleableness theorems have been exposed in M. Barr and C. Wells, Toposes, Triples and Theories."
Beck refers in the intro of his thesis to the 1966 Barr & Beck paper cited above, saying "That paper contains a summary of the present work." This might explain the later attribution to Barr & Beck jointly.