If my son lives with me half the time but his dad claims him what do I put for my taxes?

There is no such thing in the Federal tax law as 50/50, split, or joint custody.  The IRS only recognizes physical custody (which parent the child lived with the greater part, but over half, of the tax year.  That parent is the custodial parent; the other parent is the noncustodial parent.)

Who can claim the exemption and credits depends on who is the custodial parent. (By the IRS definition of custodial parent for tax purposes - this is not the same as the custody that a court might grant.).

The test that the IRS uses to determine the custodial parent is where the child lived for more than 1/2 (or greater part) of the year. The IRS will go so far as to require counting the nights spend in each household - that person is the custodial parent for tax purposes (if exactly equal and more than 183 days - The custodial parent is the parent with the highest AGI, if less than 183 days then neither parent has custody). That can usually only occur if both parents lived with the child at the same time.   And yes they are that picky.

The custodial parent may claim everything child related  UNLESS they waive the dependency exemption to the non custodial parent via a form 8332.... in that case the child may be used on 2 separate returns but only in the following way :

Only the Custodial parent can claim: (Child would be listed as non-dependent EIC & CC only)
-Head of Household 
-Earned Income Credit
-Child Care Credit

The non custodial parent can only claim: (Child would be listed as dependent)
-The Exemption
- The Child Tax Credit

See Special rule to divorced or separated parents (or parents who live apart) on page 32:
https://www.irs.gov/pub/irs-pdf/p17.pdf