Don't think you will find anything like that from the south rim. And plenty of kids do the whitewater rafting. Especially 13yo's.

Maybe go with a helicopter ride.

There are no short rides. The choice is either 7 days or 22 days. There are some using power boats that are shorter. The big problem is getting in and out of the canyons. Once in, only a couple places to get out.

There are flat water day rafting trips out of Page to Lees Ferry, but not in the GCNP, proper.

I have read you can arrange these through the Xanterra website at GCNP. http://www.nps.gov/grca/planyourvisit/one-day-river-trip.htm

The in-park raft trips are the multiple day trips that book years ahead because of the limited numbers of permits.

Some companies will pick you up at the south rim and either bus you or fly you (more scenic) to Page for the flat water day trip, then return to SR late in the day ... would be cheaper and less tiring if you were already in Page but you can do it from the south rim. Or you can drive to Page and back from SR if you have a rental car, but this makes for a long day.

Here are examples:

http://www.raftthecanyon.com/ (just the Page portion)
http://www.grandcanyon.com/coloradoriverfloattrip.html (from GC south rim, a 12 hour day with all the to/from travel ...)

Check on the age limit for the 7 y/o before you book, should be OK but possibly that's too young even for flat-water.

Like Bill_H said, you can book the trip from the Grand Canyon and they will bus you to Page, AZ to do the half day float trip on the Colorado River. This is not in the Grand Canyon, but beautiful none the less.

Unless they change the age requirements, there was a 5 year old on our float trip out of Page, AZ. He did great and really enjoyed the trip. He kept telling his Mom to take a picture of the scenery!

I'd recommend driving to Page, AZ yourself because it will be cheaper and then after the float trip you can do the 3/4 mile hike to Horseshoe Bend and see where you just floated. You could also fit in a view of Lake Powell or do the Upper Antelope Slot Canyon tour.

Utahtea

I agree with Utahtea, drive to Page and book the trip yourself. If you go the the raft the river link Bill_H gave you will find the company that we went with. They have both half and whole day trips.

I took my parents, who are in their early 60's, to GC last year at the end of June. I wanted to do a water raft trip but was afraid of what my Mom could handle so we did the smooth water raft trip. The bus left from the south rim early in the morning and we returned later that night.

I honestly thought it was the most boring thing I have ever done in my life. My parents felt the same way. It was long, hot and just really really slow. Granted I knew it was a smooth trip but I so wanted the guy to gun it a little to make some waves. It could have just not been our thing because we didn't get seats on the raft part because there were young kids and parents who snagged those seats. I won't talk about the bus driver but he's another reason why I wouldn't do this tour again.

sassy, To bad you didn't get the seats on the inflatable side of the rafts, they were GREAT and our guide did gun it right into the wake of another raft and got us wet...lots of fun on a hot day. Our 17 year old nephew was with us and he thought the float trip was the highlight of our two week trip to the national parks of the southwest.

Utahtea