The best replacement for cousin is second cousin.

The best alternatives to cousin are rated as following:
★★★★★  second cousin
★★★★  first cousin once removed
★★★★  nephew
★★★★  niece
★★★  uncle
★★★  cousin
★★  cuz
★★  cousin once removed
★★  aunt
★★  once removed
★★  1st cousin once removed
★  brother in law
★  first cousin
★  cous
★  my cousin and i
  sibling
  1st cousin
  2nd cousin
  gay marriage
  second cousin once removed
  cus

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Speaking about cousin, one probably shouldn't forget about cousin.

Other search terms related to cousin and their ratings are:
★★★★★ cousin
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Google Trends

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cousin
second cousin
first cousin once removed