I've flown several Young Eagle events, as well as Challenge Air events. One rule, every kid is different. Try to get to know them a little while walking out to the plane.
If this is their first flight in a GA plane or first flight ever, they're usually a little (or a lot) nervous. I try to judge the depth of the walk around based on their whole "attitude" towards the flight. If they seem real apprehensive, I try to get them talking by asking them about their interest in airplanes and flying but don't bombard them with a bunch of aviation terms. If they seem real responsive to all the aviation lingo, then I'll go into greater detail, etc..
Always get a picture with them. Mom/Dad/Whoever, of course, want to take lots of pics. Let them know it's okay. I'll offer everyone in the group a chance to sit in the plane, ask questions, etc. It's an experience (positive, of course) for everyone, not just the Young Eagle.
I won't go into the whole safety issues, that'll be covered in your briefing beforehand. After you're both strapped in and ready to start, I explain everything I'm doing. Sometimes, based on the kid, I'll even ask them to read the specific checklist. Get them involved.
Talk about the whole event to them. "Now, I'm going to call the ground controllers to get taxi clearance to the runway" "Okay, we're cleared for takeoff, I'm gonna taxi to the centerline of the runway", "It's gonna get a bit noisy when I push the throttle forward and you may be pressed back in your seat", "Okay, we're flying"...you get the idea.
I always, well 99.9% of the time, let them fly. The kids that were a bit nervous will just come alive right before your eyes. It's amazing. Give them a "mini" flight lesson. But, if you're a little uncomfortable with them flying the plane, don't sweat it. This should be fun for both of you!!
A couple of things to consider during these events is that you're going to be in a fairly crowded airspace. The normal traffic of the airport plus all the Young Eagle flights, especially when the first flights of the day are going out. Typically, there are two routes to fly. One route for the planes that fly under 100kts, one route for the planes that fly over 120kts and you'll have given altitudes and waypoints for each route. Not all pilots interpret the instructions the same, really keep your head on a swivel.
You're gonna enjoy the event. I guarantee it!