Can a retired woman on SS benefits claim god son as dependent if he is over 19 years old and earned about $6,000 income.

Simple answer: NO. 

For several reasons.

There are two types of dependents, "Qualifying Children"(QC) and standard ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, a relationship test and residence test. Only a QC qualifies a taxpayer for the Earned Income Credit. They are interrelated but the rules are different for each.

Your god son cannot be your Qualifying Child because he is not related. He cannot be your Qualifying Relative because he made more than $4050 of income.

The money you hear about people getting for just filing a tax return claiming kids requires you to  have some earned income (wages or self employment). Without earned income, you are not eligible for the "refundable" Earned Income Credit or Additional Child Tax Credit. Social security and other retirement income is not "earned income".

Furthermore, if your only income is Social security, you are not even required to file a tax return.

To claim him as a dependent, he would have to be related to you and because he is older than 19, he would also have to be a full time student.

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