If the custom is for a girl not to marry her cousin, does he become a mahram for her?

The woman’s mahram is
anyone whom it is permanently forbidden for her to marry because of blood
ties, breastfeeding or marriage ties. This is not based on traditions and
customs; rather it is based on sharee’ah. Allaah, may He be glorified and
exalted, has stated which women are mahrams for men, as He says
(interpretation of the meaning): 

“Forbidden to you (for
marriage) are: your mothers, your daughters, your sisters, your father’s
sisters, your mother’s sisters, your brother’s daughters, your sister’s
daughters, your foster mothers who gave you suck, your foster milk suckling
sisters, your wives’ mothers, your stepdaughters under your guardianship,
born of your wives to whom you have gone in — but there is no sin on you if
you have not gone in them (to marry their daughters), — the wives of your
sons who (spring) from your own loins, and two sisters in wedlock at the
same time, except for what has already passed; verily, Allaah is
Oft‑Forgiving, Most Merciful”

[al-Nisa’ 4:23]. 

So a woman’s mahrams on the
basis of blood ties are: her son, her father, her brother, her brother’s
son, her sister’s son, her paternal uncle and her maternal uncle. 

Her mahrams through
breastfeeding are the same. 

As for her cousin, it is
permissible for her to marry him, and it is not correct for him to be her
mahram under any circumstances, even if custom dictates that he should not
marry her.  

No one has the right to
regard as permissible that which Allaah has  forbidden, or to regard as
forbidden that which Allaah has permitted, or to claim that a cousin may
look at his female cousin or be alone with her, because that is contrary to
sharee’ah. Rather the woman must observe hijab before her cousins as she
observes hijab before all non-mahram men. 

Allaah has mentioned those
before whom it is permissible for a woman to show her adornments, and He did
not mention the cousin because the cousin is not one of her mahrams. Allaah
says (interpretation of the meaning): 

“And tell the believing
women to lower their gaze (from looking at forbidden things), and protect
their private parts (from illegal sexual acts) and not to show off their
adornment except only that which is apparent (like both eyes for necessity
to see the way, or outer palms of hands or one eye or dress like veil,
gloves, headcover, apron), and to draw their veils all over Juyoobihinna
(i.e. their bodies, faces, necks and bosoms) and not to reveal their
adornment except to their husbands, or their fathers, or their husband’s
fathers, or their sons, or their husband’s sons, or their brothers or their
brother’s sons, or their sister’s sons, or their (Muslim) women (i.e. their
sisters in Islâm), or the (female) slaves whom their right hands possess, or
old male servants who lack vigour, or small children who have no sense of
feminine sex. And let them not stamp their feet so as to reveal what they
hide of their adornment. And all of you beg Allaah to forgive you all, O
believers, that you may be successful”

[al-Noor 24:31]. 

May Allaah help us to do
that which He loves and which pleases Him. 

And Allaah knows best.