Pursuant to Florida Statute sec. 790.22, minors under 16-years old are prohibited from using, for any purpose whatsoever, BB guns, unless the minor using the BB gun is under the supervision and in the presence of an adult who is acting with the consent of the minor's parent.
Note that the FL statute says 'using" a BB gun, not possession. So it would be legal for the kid to have it but can't "use" it without adult supervision. A BB gun--by FL legal definition--is not a firearm.