My 16 year old son has earned income this year from working after school and on weekends. Do I need to claim his income on my taxes return?

No, you do No enter his earned income from work on your tax return.

He can file his own tax return and indicate on the return that he can be claimed as a dependent.

No. If he made more than $14,600, or if he had federal income taxes withheld that he'd like a refund of, then he needs to file a separate tax return.  You cannot add his wages to yours.

This article may be helpful: At What Income Does a Minor Have to File an Income Tax Return?