There is a gradual change in how the marriages between relatives were done in the Scriptures.
Cain's wife was obviously one of Adam's daughters, buch such marriages are clearly not allowed in the Scriptures thereafter.
Abraham's wife Sara was the daughter of his father, but not his mother (Gen 20:12). That is, Terah had two wives, and Abraham and Sara are children of different mothers. But later on, such alliances were not encouraged, and the law of Moses forbids it (Lev 18:11). Tamar suggested this kind of marriage to Amnon to escape from him (2 Sam 13:12-13), but such a proposal is clearly against the law.
Jacob married his mother's brother's daughter as per the instruction of his father (Gen 28:2). Leah and Rachel were his first cousins. Rebekah was the daughter of Isaac's first cousin Bethuel, who was the son of Nahor, Abraham's brother (Gen 11:29; 22:20-23). None of these marriages are forbidden in the law to my knowledge.
Othniel married his father's brother's daughter Achsah (Josh 15:15-16). She was his first cousin.
The daughter's of Zelophehad married their father's brothers' sons (Num 36:6, 11). They were their first cousins.
So, marrying first cousins is not forbidden. Neither the law, nor the New Testament forbids it.
Nevertheless, some cultures forbid. There are some societies which do not allow any marriages between cousins.
As for the customs, we cannot follow such customs which tell us to do things which are forbidden by the Scriptures. But if there are certain customs which forbid that which is allowed by the Scriptures, the best principle to be followed is in Rom 14:21, and 1 Cor 8:13, especially if those who are offended are Christians.
Again, let me tell one thing clearly, that any one who is FORBIDS marriages allowed by the Scriptures, is following the DOCTRINE OF DEVILS (1 Tim 4:1-3).