The name of the wife was not mentioned in the marriage contract and no details about her were mentioned; should the contract be renewed?

Praise be to Allah.

Firstly: 

In order for the marriage contract to be valid, both partners must be identified, and the woman must be identified by giving her name or describing her, such as saying “the youngest one” or “the oldest one” or by pointing to her if she is present at the marriage ceremony. If her wali (guardian) says, “I give this woman to you in marriage” and points to her, the marriage contract is valid. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (7/96): Among the conditions of the marriage contract being valid is that both partners be identified, because both parties to any contract must be identified, such as the buyer and seller.  Then if the woman is present and he says: “I give this woman to you in marriage,” it is valid, because pointing is sufficient for identification. If he adds more than that, such as saying, “this daughter of mine” or “So and so,” then this is confirming it. 

If she is not present and he says: “I give my daughter to you in marriage” and he does not have any other daughter, that is permissible, but if he states her name as well, this is confirming it. 

If he has two or more daughters, and he says, “I give my daughter to you in marriage,” then it is not valid unless he adds something to distinguish which daughter is meant, by stating her name or some characteristic, such as saying, “I give my oldest daughter – or my middle daughter or my youngest daughter – in marriage to you.” If he states her name as well, this is confirming it.  

You do not say whether she was present at the marriage ceremony or not, or how the marriage ceremony was done. 

Whatever the case, if the wife was present at the ceremony but he did not mention her in a way that would distinguish her from his other daughter, then the marriage contract is not valid and it must be repeated, and she must observe hijab from you until the marriage contract is done, because in that case you are a non-mahram for her. 

Secondly: 

It is valid to do the marriage contract with her father by telephone, so long as you are certain of his voice and his is certain of yours, and the witnesses are able to hear both of you. It does not matter if the witnesses are with you or with him.  

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

And Allaah knows best.