Leviticus 18:6 states, "None of you shall approach to any that is near of kin to him, to uncover their nakedness: I am the LORD."
The similar wording in Leviticus 20:17 and 20:21 show this to at least include marriage. The following verses in Leviticus eighteen give a list of those who are too close of kin to marry. They include father, mother, step-mother, sister, step-sister, grandchildren, aunt, uncle, daughter in law, and sister in law.
Technically, this does not prohibit marriage to cousins. The fuller prohibition has come partly from a desire to keep as far as possible away from something that is spoken of with such serious warning in scripture. Another reason is cultural. As our society has become more numerous and diverse, it has become more repulsive to us to marry close to family. It has also become a joke to put down those who were considered to be from the sticks--that is, rural folks who did not know any better.
But let us consider your particular situation. You refer to your child dating your third cousin. Cousins usually share a common set of grandparents (though it can be only one grandparent) and one-half of the genetic pool. Second cousins usually share a common set of great-grandparents and one-fourth of the genetic pool. Third cousins usually share a set of great-great-grandparents and one-eighth of the genetic pool. However, your child will only have one-half of your genetic pool and would only share one-sixteenth of the genetic pool with your third cousin. That is a very small amount. Many people who grow up in the same area probably have this much connection with their spouses and do not even know it.
Now, there may be other reasons that this arrangement is bad. I would certainly be very uncomfortable with a first cousin arrangement since this would be very close to the biblical prohibition. But from what you tell me about this situation, it is far from the biblical warning. We may naturally shrink from the idea of marrying a relative, but in fact we are all related. The difference is in how closely. The Bible warns against near kin marrying but the definition of near kin seems to be much more narrow than third or fourth cousins.
