This is a serious issue which I find troubling for a number of reasons. My perspective is informed by being the father of three young climbing daughters, 20+ years of climbing experience, and many years of professional guiding. I can't wait to take my girls on long routes, but will not do so without another competent adult present until I'm completely satisfied that they are ready. Any comparison between this and climbing with an inexperienced belayer in a guiding context is apples and oranges at best. All clients are either developmentally competent to provide informed consent to participate, or are accompanied by parents or legal guardians present at the belays with them. That isn't even in the same ballpark as the sort of scenario under discussion.
The most obvious problems are mechanical: Weight discrepancy, little fingers sucked in to devices, etc. I'm extremely skeptical that an 8 year old belayer is going to hold an adult taking a real leader fall in the best of circumstances. I think there is essentially no chance when dealing with a violent impact combined with their fingers being severely pinched. These issues are merely mechanical however and with thought and care the risks they pose could perhaps be mitigated to an acceptable level.
Unfortunately there are other more intractable problems. Young kids are not developmentally capable of understanding the seriousness of the situation, and should never have that level of responsibility thrust upon them. Likewise, they are typically not developmentally capable of maintaining the absolute focus and attention to detail necessary to be left unaccompanied at a belay stance off the deck. Trust me, even adult novices do astounding and foolish things at stances.
We all have to make our own decisions about what's best for our children. YMMV, but that's my perspective.
If one does choose to take a young kid up a multipitch route unaccompanied, here are some things to consider:
1. First, examine your reasons for doing it. Is it really what's best for the kid, or is it actually about your desire to climb when on kid-duty? One is ethically defensible. The other is totally irresponsible.
2. Shorten your pitches, and ALWAYS maintain direct line of sight and easy verbal communication.
3. Drag the rope up without taking a belay, clipping directionals as needed. I often feel like a novice belayer is more of a hazard than a benefit on truly easy terrain. Don't fool yourself: You're soloing anyway.
4. Choose routes with non technical descents, or be completely dialed in your technical descent skills. This doesn't mean just knowing how to rappel. Know how to competently lower from above and below, tandem rap, etc. If any of this isn't second nature to you, please reconsider your plans.
Sorry for the rant. As a father of climbing kids, this issue touches a nerve with me...