Look into Ski helmets. A good quality Ski helmet has Safety Certifications just like a DOT and Snell ratings that you have probably heard of. Child ski helmets weigh around 12 to 15 ounces. Which is much lighter then any DOT helmet.
Ski & Snow Helmet Safety Certification
ASTM F2040 is the most common snow helmet certification. This U.S.-based standard covers nonmotorized recreational snow sports including skiing and snowboarding. Look for the ASTM sticker on the inside of the helmet to ensure its certification.
(My understanding is ASTM F2040 tests are very similar to DOT certification tests)
CE EN1077 is a European certification for alpine skiing and snowboarding helmets. CE EN1077 is the most stringent snow helmet certification, by requiring not only a blunt force test, but also a penetration test (duplicating a tree branch or ski pole) Some helmets are certified to both ASTM and CE EN standards.
(I was told CE EN1077 testing is very close to what Snell requires in its certification, minus the Flame Resistance and Face Shield Impact tests.)
Check out Giro Helmet brand. They have a youth model called Slingshot. The Slingshot is both ASTM 2040 / CE EN1077 certified. This helmet is around $60.