According to his elder brother (whose real first name was a very famous Roman one), who took his stage name from a very appropriate ancient Greek god and why did people start laughing at his 1964 retirement speech?

Harpo Marx kept talking when he announced his retirement, which is the opposite of the character he usually played.

Groucho(Julius) Marx was his older brother, and the Greek God of silence, Harpocrates, was a possible source for his stage name.

https://en.wikipedia.org/wiki/Harpo_Marx

Yay PDAZ! Top Marx for a not so dumb answer!
The usual story is that he took the name Harpo due to his harp playing but elder brother Julius "Groucho" Marx said he took this god's name.
Not only was Harpocrates the god of silence, so not shouted about much, but he was often depicted as a child putting a finger to his lips.
https://en.wikipedia.org/wiki/Harpocrates

Of course, "Harpo" is also Oprah Winfrey's production company, being Oprah backwards. Isn't it fitting that the reverse of the most famous mute is the name of one of the most famous talk show hosts!