Wargames
http://www.imdb.com/title/tt0086567/

it's old (1983)
but it's light-hearted, and suitable for kids

A young computer whizz kid accidentally connects into a top secret super-computer which has complete control over the U.S. nuclear arsenal. It challenges him to a game between America and Russia, and he innocently starts the countdown to World War 3. Can he convince the computer he wanted to play a game and not the real thing?