Since you identified the person as a relative, he would not have to have been a member of the household all year (for tax purposes cousins don't count as relatives).
Assuming the relative does not qualify as a "Qualifying Child", the amount of support you provided would have to be more than the assistance, food stamps and support he got from others and himself.

See full dependent rules at <a rel="nofollow" target="_blank" href="https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Ret...>

My relative meets all of the requirements above for me to qualify her as a dependent. 

 

Could my relative lose her food stamps if I declare her as a dependent on my taxes?

If the person filling out forms with the welfare office stating that they are a separate household (even if they are living with you) then they might lose their benefits. I would ask someone who works for your local welfare office or a financial professional as the person can have legal consequences for providing improper information on their welfare paperwork. If you are not considered a member on their caseload then you should be very careful. I can't tell you 100% yes or no as I am not an employee of either IRS or welfare but from my understanding, they can lose their benefits and even face repercussions. 

Per @MinhT

#5761136

 

You can claim your relative as a dependent if he/she passes all of the following tests:

1. The person can't be your qualifying child or the qualifying child of any other taxpayer. 
2. The person must live with you all year as a member of your household (and your relationship must not violate local law). 
3. The person's gross income for the year must be less than $4,050 (Social assistance and food stamps do not count as income)
4. You must provide more than half of the person's total support for the year.
5. The person is a U.S. citizen, U.S. resident alien, U.S. national.
6. The person is not a married person who files a joint return unless that joint return is filed only to claim a refund of withheld income tax or estimated tax paid.