"Should it be acceptable for children to play their gameboys during any part of a church service?"
Here are my thoughts.
I think there is a deeper question here. How do we teach our children to engage in corporate worship?
I've been unofficially diagnosed as ADD. I can tell you that I struggled for years in a liturgical church service. Even as an adult, sitting still listening and even taking notes is difficult for me. My body needs to be doing something in order for my mind to be quiet and listen.
This begs the question, how well do you know your child? Are they the type who can sit quietly and listen to something they are really interested in or are they the type that wiggle even when watching their favorite TV programs?
If they need to be moving to stay focused, how can we constructively enable them to do what they need to do? Should we passively teach them that God made them wrong by enforcing behavior that fits into a compliance mold?
How can the church encourage its people to be what God created them to be without disrupting others' worship experience?
Here are some suggestions... if your child is old enough to be rational, discuss with them why they want to bring their gaming device. Explain that you want them to engage in what's going on. If you know they are capable of absorbing information by sitting still and listening, then that should definitely be encouraged. If they have a hard time, determine a compromise. This compromise should be something mindless and repetitive they can do while listening. Above all, don't expect more of your child than yourself. Our children notice when we whisper during church. They notice when we doodle or fidget. It's provoking them to anger to expect them to be better behaved than we are.
Here are some examples of what to do to resolve the issue. They are not meant to be taken as black and white, right and wrong, simply some things to consider.
There is a lady at my church who approached the pastor thus. She wanted to crochet during the service. Her craft is repetitive, often requiring hours of the same stitch again and again. She explained that she wasn't trying to be offensive, and honestly wanted to know if he minded. He said it didn't bother him, since he knew her heart.
I've often taken simple knitting projects to church. I find that the repetition of the same stitch again and again quiets my mind enough to listen to the service. I try not to be showy about my work. I usually work on scarves that are not too large, use materials that aren't flashy, and sit in a location that is not distracting to others.
One of my daughters draws during the service. I have talked to her about maintaining focus, and being sensitive to the speaker. She must stay quiet and she must not try to show off her work while the service is going on.
My son, who is now 16 years old used to have a hard time sitting still and listening. Now he does fine. It took him a bit longer than a lot of other kids, but he learned and eventually settled down. He doesn't doodle or fidget. He does whisper the occasional comment to his friends about what the pastor said, but I know adults who carry on constant commentary to their neighbors through the entire service.
Our 4 year old usually has a children's program to go to during the service. We allow her to play games like Angry Birds or Bubble Shoot on a tablet when there isn't an alternative for her. She's at an age where she isn't going to understand most of what is going on, so we would rather encourage her to sit quietly, than to be bored out of her mind and start getting mischievous.
So to answer the initial question: There is nothing in the Bible that says it's wrong to play games during the church service. In Hebrews 10 we read:
23 Let us hold unswervingly to the hope we profess, for He who promised is faithful. 24 And let us consider how we may spur one another on toward love and good deeds, 25 not giving up meeting together, as some are in the habit of doing, but encouraging one another—and all the more as you see the Day approaching.
In light of this, I believe parents should guide their children to be engaged in the service. They should do this by getting to know them, the way God created them, and encourage them to engage within that nature. I believe "Gameboys" and other PGD's are too mentally engaging for the majority of children to use them during a service, however I think very young children should be allowed to use them to teach them to be still and quiet while others are trying to listen. However, they should be weaned off of them as soon as they are mature enough to engage more.