The theorem often be called Lebesgue's MCT or Levi's theorem. Who did originally prove it or what is the contribution of Lebesgue and Levi respectively?
The original version of the dominated convergence, from which the monotone convergence trivially follows assuming that the limit is Lebesgue integrable, was published by Lebesgue in Leçons sur l'Intégration et la Recherche des Fonctions Primitives (1904). This is a compilation of his lectures at Collège de France over the preceeding five years.
In Sopra l'Integrazione delle Serie (in Rendiconti - Reale Istituto lombardo di scienze e lettere, 39 (1906) 775-780) Beppo Levi quotes Lebesgue's result and notes that for positive monotone sequences one can drop the a priori integrability assumption for the limit. He then applies the result to sums of positive functional series, as the title indicates.