Can I claim my 20 year old son as a dependent when he is away at college? I still support him financially.

Yes, a 20 year old full-time student can be claimed as a qualifying child dependent.   Living away at school is a "temporary absence" so you can say he lived with you all year and claim him as a dependent.  You are the one who can use the education credit on your own return, too.

You say he had income from "gigs"----do you mean he had self-employment income?  If so, he needs to file a tax return if he earned even $400 and pay self-employment tax for Social Security and Medicare.

MY DEPENDENT HAD A JOB

If your dependent has a W-2 for his after-school job, summer job, etc. you do not include the information on your own return. You can still claim your child as a dependent on your own return.  He/she can file his own return for a refund of some of his withheld wages from boxes 2 or 17 (he won’t get back anything for Social Security or Medicare), but MUST indicate on it that he can be claimed as a dependent on someone else’s return.  (Supervise this closely or prepare it for him!)

If your dependent’s earnings were over $400 and were reported on a 1099Misc or 1099NEC then he must file a return and pay self-employment tax for Social Security and Medicare.

You might also want to use free software from the IRS Free File versions:

https://apps.irs.gov/app/freeFile/

https://ttlc.intuit.com/community/self-employed/help/how-do-i-report-income-from-self-employment/00/26653

https://ttlc.intuit.com/community/self-employed/help/what-is-the-self-employment-tax/00/25922

Yes, you can claim your son as a dependent provided he meets the rules below.

• Your son must be a U.S. citizen, a U.S. national, U.S. resident, or a resident of Canada or Mexico.  There are certain exceptions for adopted children.
• You can't claim someone who is claimed as a dependent on another tax return. 
• You can't claim someone who's married and files a joint tax return. 
• They are related to you? The child can be your son, daughter, stepchild, eligible foster child, brother, sister, half-brother, half-sister, stepbrother, stepsister, adopted child or an offspring of any of them.
• Your child must be under age 19 or, if a full-time student, under age 24. There's no age limit if your child is permanently and totally disabled.
• The child must live with you for more than half the year. Living at school as a full time student does qualify.
• They may have a job, but they cannot provide more than half of their own support.

See the TurboTax article Rules for Claiming Dependents on Taxes