What is a Mennonite? Can anybody become a Mennonite? Most people who identify as "Mennonite" treat it as if it is an ethnicity, as if you are born into it. They are unaware that you cannot be a Mennonite if you are not descended from Menno Simons and/or if you do not follow his teachings.
Mennonite is not something you are born into. It is a denomination within Christianity. Anybody can become a Mennonite if they choose to. Those who call themselves Mennonite today are a hollow husk of what Mennonites of the past were. They do not understand Menno Simons' teachings.
Your last name does not determine if you are a Mennonite or not. It may determine if you are German, but it has no relation to you being a Mennonite because Mennonite is a denomination within Christianity, not an ethnicity.
Depending on the type of Mennonite, they believe you can marry your cousins, even down to second cousins. They attempt to use Abraham, Isaac, and Jacob as their proof text to support this (Gen. 24:4, 38; 28:1-2), but fail to realize that this is descriptive language and not prescriptive (a rule or direction).
They often will not marry another person unless they are Mennonite, too, even if they are a Christian. Sorry to say, but this can lead to inbreeding. God did not command Abraham to do such a thing, and just because he did so does not mean we should follow his example. Abraham had sex with another woman, Hagar, outside of his marriage to Sarah. Should we take that as prescriptive, too? Seriously. Let us think before we do things or attempt to defend them.
Any Christian denomination, such as Mennonite, that believes you should only marry people from that denomination has become cult-like. They take commands given to Israel as a nation and eisegete it for themselves (as do racists when they say we should not marry someone of a different ethnicity). Marry whomever you want, preferably in the Lord.
We need to stop holding on to blind traditions that are the inventions of men. Cling to the New Covenant Scriptures and understand your new identity in Christ Jesus our King.