If you can qualify to claim him as a qualifying relative it's possible for you to claim him.  The rules are placed below for your review.

It depends. if she meets the Qualifying Relative Rules for you as shown below, the answer is yes.  

• The two most important rules for you are the taxable income on her return and whether you provided more than half of her support. Click the image attached to use as a guide for determining support if you meet the gross taxable income test.

Qualifying Relative:

• You provided more than half of their support.

• They made less than $4,050 in gross taxable income. (Social Security income generally doesn’t count here.)

• They live with you (365 days in the year) or they are related to you.

• They aren’t a dependent on someone else’s taxes.

• They aren’t doing their taxes with a spouse (married filing jointly).

• They are a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.

  You can’t claim a dependent if you are a dependent on someone else’s taxes.  Use the link here for more information.  Rules for Claiming a Dependent on Your Tax Return

• IRS Worksheet to Determine Support (click this link to print the worksheet)

Would I have to pay a penalty if he didn't have insurance for the year?

@sullivan It's possible.  Here is a link for possible exemption from the penalty if it should kick in for your return.
<a rel="nofollow" target="_blank" href="https://www.irs.gov/affordable-care-act/individuals-and-families/aca-individual-shared-responsibilit...>

Yes, if that is his only income. The income is below the $4050 limit. That assumes you meet the other rules; he lived with you all year and you provide most of his support.

He will have to file a tax return of his own. He will not claim his own exemption ("claim himself"), since you will be doing that. He will owe no income tax, but must pay social security and medicare tax on his net earnings.

@Hal_Al Thank you for the additional information about the dependent's tax return.