It was named Pasquino because the statue was placed near a tailor's shop named, of course, Pasquino.
On it the Romans placed notes of political discontent since it was illegal to outwardly voice resentment to the government. When you visit the statue a few blocks from Piazza Navona you'll even see notes posted today. They'll be in Italian so take a few pictures and translate the notes when you get home or ask a nearby local if they would translate for you.
One of the more famous notes was placed on Pasquino in the mid 1600's directed at Pope Innocent X. This is the pope that funded and built the famous fountain of Four Rivers by Bernini in Piazza Navona with taxation of the Romans. The note read, "We don't want obelisks and fountains; we want bread, bread, bread."
A little on the story of Patroclus and Menelaos. Achilles chose to remove himself from the war against the Trojans in the Illiad due to a slight by general Agamemnon. The slight was over a woman who Achilles claimed as his own as a spoil of war. General Agamemnon took her from Achilles. Achilles removed himself from the war and remained in camp and vowed not to fight until he got back what was rightfully his. His cousin Patroclus wanted to join the fight, but Achilles refused.
More on the story of Troy and the importance of the Illiad to art in Italy and Greece.
Oh, one more thing, the Borghese museum statues are organized starting with the statue that ultimately kicked off the Trojan war with Venus Victrix and ending with Aenaes fleeing from Troy to found Rome.