We absolutely love our Islabike Beinn 20 Small. The geometry, quality and features are head and shoulders above anything else currently in the market. My daughter has also ridden Specialized, Giant, and Trek bikes and there's absolutely no comparison in terms of her comfort level with the bike and her speed. For reference, she's 5 years old and recently completed a kid's triathlon and did the 4 mile bike portion on her Islabike at an average speed of 14 mph.

If you're planning to buy new anyway and it's going to be passed down, I would spend the money on an Islabike. We were a little concerned at the outset about how she would adapt to the gears and the handbrakes. Initially, she spend most of her time on gear 3 or 4 and wouldn't change, but after 6 months she was pretty adept with the gears. The handbrakes were no problem either--after a day, she was a pro at stopping.

If you really don't want the hassle of gears, I'd look at the new Cleary bikes Owl, which will start shipping in late August: Owl - Cleary Blue - 20" Singlespeed Bike | Cleary Bikes
At 17lbs, it weighs in a full 7 lbs lighter than any of the non-racing 20" bikes other than Islabikes. The one caveat is that it doesn't have a coaster brake.

However, I'm very much anti-coaster brake. I think they're much more of a hindrance than a help. There's nothing like standing up to pedal, accidentally putting your weight a little too far back, and coming to a screeching halt. They also stop kids from getting into "power pedal position" to accelerate from a stop. I dislike them so much that we're springing for a Cleary bikes Hedgehog for my son's 4th birthday, because it's the only bike in the U.S. that doesn't have a coaster brake on the 16".

If you're buying used, my recommendation would be a Specialized Hot Rock, because they're marginally lighter than other brands.