This theorem has been proven by many people and in the context of many diverse fields of mathematics since Lagrange presented "Nouvelle methode pour resoudre les equations litterales par le moyen des series."
It is so ubiiquitous that I've often thought that getting all the proofs together in one place would capture in a unique and compelling way the ebb and flow of the styles, habits, aesthetics and interests of the mathematics community over time. Indeed, I started on this project and quickly discovered that it was more than I could possibly do on my own. Just starting the project however gave me an even deeper appreciation of works such as Dickson's History of the Theory of Numbers and Fletcher's Index of Mathematical Tables.
I've got a list of 50+ proofs, a large collection of books, papers and articles and a bibtex bibliography if anybody is intrigued.
IMHO, one of the most general statements is Krattenhaler's "Operator Methods and Lagrange Inversion: A Unified Approach to Lagrange Formulas." One of the most unique is Abdesselam's "A Physicist's Proof of the Lagrange-Good Multivariable Inversion Formula."
Cheers, Scott (sbg at acw dot com)