my cousin comes here from Mexico on a travel 6 month visa how can she get a green card to stay permanently?

The short answer is: she can't.
The long(er) answer is: if she can show a good reason to apply for a green card then she may be allowed to stay.

Usually you get a green card in one of the following ways:
1) Marriage to a US citizen or an existing green card holder (but usually a US citizen)
2) Employment, where a US employer sponsors you to get a green card
3) The green card lottery

There are a couple other ways (such as if you invest $500,000 or more in a business resulting in the employment of a number of US citizens) but I doubt she would qualify for those anyway.

Not possible. Tourist visas cannot be converted into any form of temporary residence or immigration visa. Just applying for change of status is prima facie evidence of immigration fraud on the tourist visa application. $10,000 fine and permanent bar to getting any future visa.

She has no choice but to go home for at least 6-12 months, then see if she qualifies to get a job in a category for a temporary employment visa. She'll need at least a college degree and 2 years post-baccalaureate experience to apply for a work visa. The job market is very bad, so unless she has special skills, there isn't much hope of a work visa for at least the next 12 months any way.

If she wants to immigrate and get a green card, she has to apply in her home country. And she has to have a sponsor who is an immediate blood relative (not a cousin). If she is accepted, she will have to wait until her priority number comes up. It will take several years and she cannot enter the US until she gets permission to enter on the immigration visa.

It's not that easy. I agree with Soda Popinski, those items he lists are her options. Good luck.