Ship legally possessed firearms through a properly licensed gun dealer. Do this only after determining that the recipient can legally possess the weapon and has complied with the laws of his particular state.
You can ship a handgun through the U.S. Postal Service if you are an FFL holder, if you are not you need to use UPS or Fedex. For Fedex it needs to be shipped to an FFL holder, if your cousin doesn't hold an FFL ship it to a local dealer in his or her area who will transfer it for a fee of around $30. And I think for UPS you need to affix a UPS label requesting an adult signature, and you may need to be licensed I'm not 100% on that. And usually for handguns you need to ship overnight.