Is it a sin for a father and his sons to use their private home to open a project they had planned beforehand, after the father had promised a female relative to rent the house for the same purpose and then went back on his word?

Since the father conditioned his response on God's will, he did not break his promise by backing out of leasing the shop to the woman. There is no objection to using the shop for the same purpose, because her mere thought of establishing the project does not grant her an exclusive right to it. However, she should be appeased and maintained good ties with, as she is a relative by kinship.