Is the objection of the family to their daughter marrying a divorced man who has a son, and the mother's condition that the husband provide an apartment in the daughter's country, a valid stance according to Sharia? And what should the daughter do in this regard?

If the suitor is a man of religion and good character, and you desire to marry him, then your family does not have the right to prevent you. You may stipulate that he reside with you in your country. If you agree to reside with him in his country, then your mother does not have the right to force you to demand a residence in your country. We advise you to try to convince your mother to agree to the marriage without making it difficult for the suitor by demanding a residence in a country other than his own. Seek the help of some relatives or those who have influence with her, and strive in supplication. You must honor your mother and be dutiful to her, and not mistreat her or fall short in her rights, for her right is great. Before proceeding with the matter, consult the wise among your family and seek guidance from Allah through Istikhara.