The "Lego Batman" series of video games is very similar to the Lego animated movies in tone and style, and the games are rated "E" by the ESRB. "Scribblenauts Unmasked: A DC Comics Adventure" also features Batman as a central character and is appropriate for young children.
There are three games in the "Lego Batman" series, as of March 2015. "Lego Batman 3: Beyond Gotham" is the most recent, and it is available for all of the major consoles of the current generation, as well as the Playstation 3, Xbox 360, Playstation Vita and Nintendo 3DS. There is also a PC version available for Microsoft Windows. The previous games are available for older generations of consoles and handhelds, but both also have PC versions.
"Scribblenauts Unmasked" casts the player as a boy named Maxwell who uses a magical notepad, but he spends much of the game working out of the Batcave with assistance from Batman. Maxwell also uses a utility belt filled with gadgets similar to the one Batman uses.
Most other modern Batman video games are pitched more toward adults. The "Arkham" series published by Warner Bros. is very popular, but may be too violent, frightening or confusing for young children.