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1.
A time of morning (or night), usually between 3AM - 6AM in the local time zone that requires a person to interrupt normal sleep or waking patterns for often dubious reasons. Often used in reference to extremely early arrivals to airports, or to join business teleconference sessions with participants in multiple time zones. Also see stupid o'clock.
"I don't look so good because I had to be at work at silly o'clock this morning to join a videocon with our development office in Greece."
by ToddUncommon September 12, 2008
 
2.
A time so late in the night where everything is funny to you and your friends. Usually achieved after being awake for 20 or more hours. Similar to being high, even the least funny things appear hilarious.
"Knock knock."
"Who's there?"
"You're gay."
"Hahahahahahaha! That wasn't even funny... it must be silly o'clock."
by Baron Von Yiffington August 21, 2009