Son of Perdition

What does Jesus mean by, "the son of perdition”?
απωλειας (apoleias), the original Greek word translated "of perdition," means "of destruction" or "of ruin," so "the son of perdition" means the son of destruction or ruin.

Who is the “lost... son of perdition”?
Judas Iscariot, who, after betraying Jesus, committed suicide instead of repenting, and went to hell for everlasting destruction.

By saying, "Those whom You gave Me, I have kept; and none of them is lost except the son of perdition" (John 17:12), is Jesus saying that He lost someone that He was expected to keep or should have kept?
No, "gave" does not connote the need to 'keep.' When chosen as one of the twelve apostles of Jesus, Judas Iscariot may or may not have known that he would betray Jesus three years later. Satan may have thought that he had managed to infiltrate Jesus' inner circle with a thief whose love of money could be used later to betray. But God the Father and Jesus knew perfectly well that Judas Iscariot would betray and that his betrayal would lead to Jesus' arrest and crucifixion, and had included him among the twelve for that purpose. Throughout the Bible, God used both heaven-bound and hell-bound people to direct history. Those who associate being used by God with salvation while continuing to live in sin do so at their own peril: "Many will say to Me in that day, ‘Lord, Lord, have we not prophesied in Your name, cast out demons in Your name, and done many wonders in Your name?’ And then I will declare to them, ‘I never knew you; depart from Me, you who practice lawlessness!’" (Matthew 7:22-23)

Did Jesus let the eleven apostles hear this prayer to God the Father?
Yes: "these things I speak in the world" (John 17:13).

Why?
"That they may have My joy fulfilled in themselves" (John 17:13), and also for John to record this prayer in his gospel.