Can I claim the $500/mo I pay my son to live with and take care of me?

No.  If you are paying your son to live with him, that would be rent.  Rent is not deductible.  

Q. Can I claim the $500/mo I pay my son to live with me, in my home,  and take care of me?

A. Simple answer: No. 

If the care qualified as "medical care", then you might be able to claim a medical deduction. But medical deductions are only an itemized deduction and even then you can only deduct the amount (of your total medical) that exceeds 7.5% of your AGI.  

If you did deduct it, your son would have to report it as income and pay income tax, social security tax and medicare tax.