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.
Islamically, if a Muslim wants to undertake a particular matter that is permissible, he should perform Istikhaarah (guidance prayer). In fact, Istikhaarah is to put the matter in the Hands of Allaah The Almighty so that He will choose for His Slave what is better for him, as it is Allaah The Almighty Who knows the outcome of the matters and He is Omnipotent over everything. This meaning is indeed clear from the wording of the Istikhaarah. Thus, a Muslim should not determine in his supplication a specific choice, otherwise, this cannot be considered an Istikhaarah but just a supplication.
Secondly, the result of the Istikhaarah is not necessarily that a person sees something (in his dream), but the most important result of the Istikhaarah is whether a person is enabled to do a given matter or not.
Thirdly, if this young man is not religious and does not have good morals, then you should not perform Istikhaarah to marry him; Istikhaarah should only be performed for permissible matters but not for what is forbidden or disliked; this is clarified in Fatwa 88646.
Fourthly, it is permissible to supplicate with this condition: “O Allaah make my cousin a religious man and bless me with him in the manner that You are pleased with.”
However, it appears from the words of his mother – i.e., your aunt – that problems might result from this marriage, so we fear that this would lead to the cutting of ties of kinship. Therefore, it is better for you to absolutely forget him and look for another man to marry.
Indeed, it is permissible for a woman to look for a pious husband provided that she observes the Islamic etiquettes in doing so.
For more benefit, please refer to Fatwa 82471.
And Allaah Knows best.