How long does a child have to live with you to claim them?

Generally, the child must have lived with you for more than half the year. If a child was born or died during the year and lived with you (or in the hospital) while alive, you can treat the child as having lived with you all year. If the child was kidnapped, you can treat the child as still living with you.

If the parents were divorced or separated, then the rules take several pages to explain. If you are really interested, see page 13 of Publication 501:

http://www.irs.gov/pub/irs-pdf/p501.pdf