All of these other "contest" tricks scare me. Here's a tried-and-true old trick easily accomplished by anyone who can "sleep."
I did this animation several years ago for a yo-yo retail site, drawing in 8-bit black & white like our cave-dwelling ancestors did.
A quick yo-yo-bio: I taught a yo-yo "workshop" once a week, during the Great Ball Bearing Revolution of 1997. Lots of kids were using those awful "auto-return" contraptions that are a heinous obstacle to skill development... but I was preaching the gospel of "wood" and ball bearings, and I daresay that I rescued a significant number of impressionable youths from the sinister crutch of auto-return. Who knows, perhaps one of them has entered this yo-yo contest? The hardest trick I can do is the "rock the baby inside the Eiffel tower..." I can do the Brain Twister, the "star," that kind of thing. But my favorite thing to do with a yo-yo is looping all day long. I usually knock myself unconscious when I try to "Shoot the Moon." Favorite yo-yo? Spintastics Tornado II, Tom Kuhn's Roller Woody. Favorite string: by Albion Manufacturing, now they're apparently "Tassels.com," and I don't know if they still make string (luckily they sent me a big envelope full back then, and I've still got lots!).
Now that all of that is out of the way, here's how to rock the baby:
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Step 1: Sleep, Make a "C," Grab a Loop, Make a Triangle, Rock, Get Out
(I consolidated the steps, because they're all pretty clear if you watch the animation)
1. Put your yo-yo to sleep, with a nice hard spin so it stays sleepy for a long time. Get it spinning at a perfect vertical, or you'll reduce it's sleep time, and make the rocking part more difficult.
NOTE: With this and ANY "string" trick, first practice with a "dead" yo-yo. No pressure that way. When you've got it down, then sleep it.
2. Hold your yo-yo hand out in front of you. Make a "C" shape (or "half moon" or "half circle..." you get the picture) with your string finger and thumb, so that your thumb is between you and the string. (The animation shows a left-handed yo-yo-er, I have no idea why)
3. With your non-yo-yo hand, pull the string toward you, using all four fingers. Don't pinch the string; let it slide over your fingers as you pull.
4. Push the string forward and under the yo-yo, spreading it into a triangle (pinky & index finger are the "points"). The bigger the triangle, the easier the trick.
5. Rock the yo-yo back and forth through the triangle.
7. Get out: either "throw the baby out of the cradle" by looping, or just drop the string and snap it all back into your hand.
8. Revel in your accomplishment, and the knowledge that you have taken your rightful place in the Grand Continuum of Yo-Yo-ists, which includes Tommy Smothers and some ancient Greek enthusiast and an extremely intimidating Phillipine warrior ("Drop your sword or I'll walk the dog at you!!"). Congratulations.