Hi, Piera! All signature restaurants at the WDW resort require two table-service credits from the 2010 Deluxe and basic Disney Dining Plans. However, if you've purchased the 2010 Premium or Platinum Dining Plans, they already include signature dining and will only take one table-service credit.
The Hollywood Brown Derby in Disney's Hollywood Studios is a signature restaurant, and it does have a children's menu. At the time of this posting, the children's menu includes such items as chicken noodle soup and chopped salad for appetizers; grilled cheese, meaty macaroni or a hot dog for an entree; and specialty desserts for kids. If you are on the basic or deluxe dining plans, I would consider not using the two table-service credits for your child(ren) and just paying out of pocket. The prices (nor food selection) on the kids' menu is extraordinary, and I wouldn't want to waste two credits on an experience that they may or may not appreciate.
Also note that there is an expectation of a "resort casual" dress code for signature restaurants which means that cut-off jeans, tank tops and gentlemen's hats are not acceptable in the dining room. Enjoy your meal at the Derby, and make sure you keep an eye out for anyone famous! Have a wonderful day and a magical vacation!