@coffeecupsandfairylights raises good points!
I'm a fan of standards though having grown up with them. They're great with children, gentle, fun loving - wonderful temperaments and have a natural sense of humour.
All poodles are incredibly intelligent so worth bearing in mind that some might be harder to train than others, simply because they're more independently minded!
Although I believe the difference in between size variations is less in poodles than other breeds, do you research as to temperament and behavioral variations between standards and miniatures. Miniatures are more likely to suffer from hip issues than standards - standards poodles are physically more robust than miniatures and toys and actually a breed that isn't inclined to much in the way of physical impairments/disabilities at all.
We got our most independently minded standard when I was the same and as your children. He took a lot of training in terms of recall when off the lead but strength wise no problem for me to manage. He was angel in all other respects.
I also think that training techniques and leads have evolved a lot since then so a pulling dog is much easier to manage - definitely get a harness rather than attach a lead to the collar.
I'd absolutely opt for a standard over a miniature no matter what age the children were :)