Coloring books for adults is an offspring of art therapy which is a mental health profession that is the process of making and creating artwork which intends to "explore feelings, reconcile emotional conflicts, foster self-awareness, manage behavior and addictions, develop social skills, improve reality orientation, reduce anxiety and increase self-esteem" (according to the American Art Therapy Association). So basically, it's similar to good old therapy. Also, art therapy is not only about mental health and improving yourself - it's a form of personal expression, as well.
There may be many fun games for kids over internet but coloring pages from Disney story books have no substitute. They can be provided by school teachers as a fun activity too. But you also need to award your child when he is done with the coloring. Kids need appreciation and it can be done by encouraging them through rewards.
According to Wikipedia: A coloring book (or colouring book) is a type of book containing line art for a reader to add color using crayons, colored pencils, marker pens, paint or other artistic media. Coloring books are generally used by children, though coloring books for adults are also available. They cover a wide range of subjects, from simple stories for children to intricate or abstract designs for adults. The golden age of coloring books is considered to be the 1960s.