All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, please accept our condolences on the death of your father, “Inna lillaahi wa-inna ilayhi raaji’oon", (To Allaah we belong and unto Him is our return). Everything with Allaah has an appointed time, so you should be patient and seek the reward from Allaah. We ask Him to forgive your father and increase your reward.
It is permissible for your mother to go out during daytime in order to get her pictures taken for issuing her passport.
Getting the picture of a woman taken for a necessity, like issuing a passport, -even if she is in her waiting period- is permissible. Additionally, it is permissible for her to unveil her face for this purpose, as she is forced to do so. Allaah Says (what means): {But whoever is forced [by necessity], neither desiring [it] nor transgressing [its limit], there is no sin upon him. Truly, Allaah is Oft-Forgiving, Most Merciful.}[Quran 2:173]
However, it is a woman who should take her picture, and it is only when there is no woman to take her picture that she is permitted to uncover her face in front of a man for this purpose.
Furthermore, according to the view of the jurists it is permissible for a woman who is in her waiting period to go out for a need. However, if it is possible to delay taking the pictures until her waiting period expires, then it would be better, because in principle a woman who is in her waiting period is obliged to stay in her home and not go out except for a need.
For more benefit, please refer to Fataawa 88684 and 82015.
Allaah Knows best.