immigration
I'm pretty sure there is no preference category for cousins.
The most direct route would be to get a green card for your wife. When she becomes a citizen, in about 5 years, she can get a green card for her mom or dad - who should get citizenship in about five years - who should get a green card for their sibling - could be a long wait. Then that sibling (by now it should be the mother/father of the cousin) should get their citizenship (5 years) and then they can apply for their child - the original cousin in question. All the relationships have to be verified and documented from a VERY early age, if not at birth, and if through the father, marriages should have taken place.
Very indirect route and takes a long time.
If the INS knows she's the last cousin, they won't give her a visitor's visa because they know she is liable to over-stay...Now, if she was able to get to NY, a job as a cook in a Dominican restaurant could get her a 6th preference - as far as I can remember.
A good immigration lawyer would be invaluable in that situation.
I am not an immigration lawyer, but after staying at a holiday inn express last night, ...
Good luck to your family and your wife's cousin.
mk