My son has been using a Schwinn similar to the Gremlin for a few years and we've been really happy with it. I was attracted to it by the full chain case-- didn't want his shoe laces or fingers getting in there. Make no mistake, it's really heavy for the size. We bought it new on a good sale at Performance but in retrospect, I should've been patient and gone with Craigslist which is what I'm doing now that he needs a 20".
Santa is bringing my daughter (who is 5 years old with an 19.5 inch inseam) the Beinn 20 small for Christmas. It is a gorgeous bike. The gear shifting is smooth, the brake levers have a light touch. It weighs 17.3 pounds, which is about 2 pounds LESS than her current 16" Hot Rock.
My second choice would be a Specialized Hot Rock 16"
We had the Novara 12" Stinger and that sucker is a brick. We also had a Trek Mystic 16" (the girl's version of the Jet) and it was also very heavy and has, in my opinion, awkward geometry. The handlebars sit up very high, which puts the center of mass very high and makes it harder to handle. My daughter learned to "ride" without traning wheels on the 12" Stinger, but it was basically to ride short distances or ride around in circles. She outgrew it but wasn't able to handle the 16" Mystic without difficulty, which discouraged her from riding. It wasn't until we got a 16" Hot Rock that she was really able to ride. Now she regularly rides up to 4 miles on her bike.