Zblip, you won't want to use your bike to get around Manhattan at all, that's what taxis, the subway and your feet are for. I used to commute into the city from Jersey on my bike and parked at the garage previously mentioned, on 43rd, between 5th and 6th. I'd still ride every day but life circumstances aren't letting me do that anymore.
My suggestions:
1) Ride your bike to the city and park it for the duration of your stay.
2) I suggest you take the long way to NYC and use all the beautiful back roads to the west of I-87 (Route 30 down through the Adirondacks to Hancock in New York is a gem) or east of I-87. It will take you a little longer but you'll have more fun.
3) There are motorcycle friendly parking lots in and out of the city. A good internet search will turn them up for you. Here's a page that at the bottom lists lots that take bikes. I parked at ParkFast in the city and they are a VERY well run operation:
Motorcycle parking in Manhattan & NYC motorcycle parking
4) Stay in the city, NYC never sleeps and you and your son will want to enjoy every minute of it. The hotel prices are always outrageous but a search will probably turn up something acceptable in your price range. My motto: all hotel rooms look the same while you sleep.
5) When you ride into the city, have your directions/map in paper form, GPS gets worthless and funky amongst the tall buildings so don't count on it to be accurate.
6) Even at rush hour, traffic, at least to me, isn't that bad or dangerous, but hey, I live here and am used to it. You certainly have to be patient and gotta pay attention. The previous commenter saying to be aggressive is oddly correct. I know you're worried about your son, as I would be. Plan to arrive and depart late morning, the traffic is usually light then, particularly at the tunnels and bridges into Manhattan.
7) If you have a smart phone, you can use google maps or one of many available apps to help you get around on the subway, which will save you a ton over cabs. If you get confused in the subway just ask somebody how to get to where you are supposed to go, despite the movies, New Yorkers are friendly and helpful. If you get lost don't worry, the trains only go two ways and the adventure is half the fun.
8) HAVE FUN!!! I bet your son will be thrilled with the trip.