If you do not want a litter of new mice every few weeks, then either get a single mouse or two female mice from the same litter. Mice may be a little more aggressive with their biting habits, but should become much more owner-friendly after a period of adjustment. Much like guinea pig ownership, but with a little more maintenance. Rabbits function best with other rabbits, and tend to need more room than a similar sized guinea pig. You might consider constructing an outside rabbit hutch, or at least an penned area for exercise purposes.
Rabbits are very docile, and can withstand some fairly rough handling by a new owner.
They are vegetarians, so they rarely develop a taste for owner finger. However, they do leave a substantial amount of feces throughout the day, and cleanup is a daily concern. Children may enjoy rabbits as playmates, but the main caretaker should probably be a responsible adult. Great introduction to the reptilian pet world, without the stigma sometimes ascribed to snakes and iguanas by non-owners. Turtles can be kept in small terrariums or fishtanks, and are very self-sufficient.