As a newly enlisted unmarried soldier (without children), how do I add my mother as a dependent?

PVT (Join to see) To qualify as a secondary dependent, your mothers income, not including your contribution, must be less than one-half of the actual living expenses. The law requires the individual to be ‘in fact’ dependent on you, the service member. Your contribution must be more than one-half of the dependent’s actual monthly living expenses. Documentation to prove living expenses and your contribution must ALWAYS be provided.

And this is HOW you do it if eligible
https://www.dfas.mil/militarymembers/SecondaryDependency/secondarydependency_Army.html

When you get to Basic Training, during your in-processing, you will hit the DEERS section. You can add your mother as a dependent at that time.

You need to complete a secondary dependent packet and submit it to DFAS. Once DFAS approves it, then you will need to enroll your mother in DEERS and in Tricare.