How can I become a Mexican citizen through my mother?
If your mother is Mexican, you are already a Mexican citizen.
If your mother was a citizen of Mexico at the time of your birth, you ARE a Mexican citizen. You need your birth certificate - proof you are her child - and proof she was a Mexican citizen at the time you were born.
You were born as a Mexican citizen if one of your parents was a Mexican citizen at the time of your birth. The principle for this is jus sanguinis, and it's the primary way to become a citizen of a country, any country, and every single country on this planet accepts it. So grab your mommy and head over to the Mexican consulate where she'll have to prove that she was a Mexican citizen at the time of your birth, and that you are her biological son. For the latter one, your birth certificate listing her as your mother should be sufficient.
If your mother was a Mexican citizen at the time of your birth you're likely already considered a Mexican citizen and should contact the nearest Mexican Consulate to find out how to obtain your Mexican passport.
If your mother was Mexican when you were born, you already are Mexican, just apply for a Mexican passport