Ah, good old Eugenio, the all-star of European servants. Sure, there are tons of servants in this book, but Eugenio gets called from Paris to Italy just to be sure that Milly Theale has the very best of caretakers for her final months. Eugenio is an especially interesting dude in this book because he seems to be even more cultured than most of the rich people. As the book tells us:
The great Eugenio, recommended by grand-dukes and Americans, had entered her service during the last hours of all—had crossed from Paris, after multiple pourparlers with Mrs. Stringham, to whom she had allowed more than ever a free hand […] and had dedicated to her […] all the treasures of his experience. (220.127.116.11)
On top of this, Eugenio takes a very special interest in Milly Theale. He even becomes downright protective of her when Merton Densher starts showing up for visits. Merton knows almost right away, for example, that "Eugenio took a vulgar view of him, which was at the same time a view he was definitely hindered from preventing" (18.104.22.168). Eugenio thinks Merton is after Milly's money. And yeah, he sort of is.
Eugenio functions as a grim gatekeeper toward the end of this book. If you want to get to Milly, you have to go through him.