What is better for my 10 year old overweight child: a treadmill or a cycling machine?

PlAy is better. Active lifestyle is what i stress and for a 10 year old i would use the word play, kicking a soccer ball, jumping rope playing tag playing hide and seek, riding a bike also chores like walking your dog yard work cleaning your room. I would not say treadmill or jogging! make it fun.
Yuck! Neither! unless your child is super motivated, they'll soon tire of either of these boring pieces of equipment. Sports are much more fun. Healthy competition. Exposure to peers. Exercise that has an immediate goal and lots of feedback. Find something your 10 year old enjoys, even if its just ultimate frisbee. Softball, soccer, hockey, football, bowling, tennis, basketball - something!