Resources on Italian Game, Two Knights Defense with 4. d3 h6 and quick ...g5 Do any books or other resources discuss this variation for Black? Does it have a name?
The general idea is to go 4...h6, then ...d6 and ...g5 in some order, depending on what White does.
S. Mamedyarov plays this line a lot.
Sample line:
 [FEN &quot;&quot;]
    1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. d3 h6 5. c3 d6 6. O-O g5
FM Andrey Terekhov's Two Knights Defense repertoire on Chessable uses this variation against 4.d3.
That repertoire deserves to be more widely known: it's one of the best on Chessable, the author updates it frequently, and it's free!
In the introductory text of 4.d3 h6 he notes:

In the beginning, this line has been mostly used as a surprise weapon, but in the past two years it developed its own body of theory. GM Alexey Kuzmin published a couple of articles summarizing the latest developments in this line in &quot;ChessBase Magazine&quot; (in 2018) and in Russian &quot;64&quot; (2019) .

So those are sources too.
And according to a ChessPub discussion, there are also two analyzed games in this line in Alexander Ipatov: Unconventional Approaches to Modern Chess Volume 1; Thinkers Publishing 2019, although I don't see it in that book's list of contents.