Closing a letter from two people
Well it depends on who you’re writing to in the family and the actual relationship you have with that person, but simply using names feels rude and inappropriate in letters in most cases. Generally it is less inappropriate if you go more polite and make the recipient feel close with you in writing. And I think the reason is because Chinese like to establish relationships with other people. Even for people not in the same family, we want to call them by 哥、姐、弟、妹 to make us feel more connected/close, not to mention that we have already have connections by family. 

For the same generation, I feel it should be okay just use your name when you write to your cousins or siblings, depends on your closeness with that person. If you only use your name, it is better you start with that person’s relationship at the beginning of the letter, for example 表姐、表哥、堂弟、堂妹 etc, then in the end you can probably close with your name. And if you want to start with that person’s name only without indication of the relationship, normally it is writing to someone younger than you, such as 弟弟、表妹 etc.. With some indication of relationship at the beginning you can close the letter with your name. It feels very distant if you don’t use some word to indicate your relationship with that person in writing. 

If you write to your spouse it depends on how you call each other. In the old days people use 妻 xx、夫 xx etc but i would imagine people use nicknames more nowadays.

If you write to parents, normally you use 儿xx、子 xx、女xx 、儿媳 xx etc in conjunction of your name without/with surname(generally without). If you write to your parents’ siblings, it’s very inappropriate not to use relationship words like 姨、叔、伯 etc unless you have a very close relationship and you have a special nickname between two of you, which is hard to imagine for me.

For other relatives, you use the relationship in front of your name without/with surname. For example, 侄 xx、侄女 xx、外甥 xx、外甥女 xx etc

If you write to grandparents, you call them by 爷爷、奶奶、外公、外婆 etc with no name mention and you close by 孙子xx、孙女xx etc. The name can be omitted if you're the only grandchild in the family. There’s less room to argue here.