Should he still do the marriage contract when the girl’s father has died and her mother is still in mourning?

Praise be to Allah.

It is permissible to do the marriage contract with this girl even if her mother is still in the ‘iddah period following her husband’s death, because there is no shar’i reason to prevent that, and it is not essential for the marriage contract to be done that there be a celebration or party. It is sufficient for the wali (guardian) and two witnesses to be present. But you must pay attention to the feelings of the girls and her family. If this will not upset them or it will detract from their joy, then praise be to Allaah. Otherwise, you should delay the marriage contract until their grief at what has befallen them has abated, and the mother is able to wear her adornment at her daughter’s wedding if she wants to do that. It is well known that during the ‘iddah following her husband’s death, a woman is not allowed to wear jewellery and adorn herself or wear perfume. Please see the answer to question no. 13966. 

To sum up: There is no impediment in terms of shar’i rulings to doing the marriage contract in such circumstances, if the girl and her family agree to that, and it will not cause them any harm or upset them. 

And Allaah knows best.