https://www.irs.gov/taxtopics/tc553
This seems to suggest that I can file them on my tax return, including capital gains: "If your child's only income is interest and dividend income (including capital gain distributions) and totals less than $13,500, you may be able to elect to include that income on your return rather than file a return for your child. See Form 8814, Parents' Election To Report Child's Interest and Dividends. "
