What documents would my partner need so we could get married if he is undocumented from Mexico?

To get married in New York, your partner needs nothing but his passport - the same as since June 24, 2011. But the problem is not getting a marriage license: the problem is getting a legal immigrant status. First, there is little doubt that USCIS will take time to bring its procedures in compliance with the Supreme Court's decision in Obergefell v. Hodges. Until it happens, answering your question is going to be a bit speculative. However, you can be certain that your marriage is not going to instantly resolve your partner's immigration problem. The law is clear: the alien who was admitted into the U.S. and later married a U.S. citizen, can apply for adjustment of status to that of a permanent resident if the U.S. spouse files an immigrant petition for the alien spouse (and other required documents). This rule works even if the alien overstayed his/her visa for years. However, if the alien entered the U.S. illegally, he/she is not eligible for a green card even through a marriage to a U.S. citizen. One cannot adjust status if he has no status. The alien must leave the U.S. and wait (usually, 10 years) before applying for a permission to re-enter the U.S. There are exceptions to this requirement, and there are procedures for obtaining an exception. And we have no reason to think that these procedures for same-sex families will be any less difficult than they are for the rest of the families having an illegal alien as one of the spouses. In 2011, the New York Legislature cleared away many hurdles and indignities that were in a way of your normal family life together. Now, the Supreme Court said that your family is legally equal to the traditional families. If you love this man and wish to marry him, by all means, do so. But talk to an immigration attorney before you file any papers with USCIS for your spouse. Remember that doing so gives notice of your partner's illegal presence in the U.S. to the authorities. No matter what enforcement policies are currently in effect, they can change any day (and will very likely change at the beginning of 2017). Before sticking his neck out, you both should have a clear notion of what the prospects are for getting a legal status for him, what you must do to achieve this goal, what are the odds of success, and what would you do if it all fails.

To marry is one thing; to get an immigration benefit is something completely different. If he entered the US unlawfully, he is not eligible for legalization, even if married to a US citizen, under the current immigration laws. That would be the same regardless of whether it is a same sex marriage or not. Please let me know if you have any additional questions or how I can be of further assistance to you in this matter.