What is the ruling of Islamic law regarding a father and brother refusing to reinstate a son as a partner in their business after his private venture failed, knowing that he had worked with them for 20 years, and given that the eldest brother threatened to quit if the son returned as a partner, and the father offered the son support for a new project or a sum of money to forgo working with them?

Your work with your father in trade does not fall outside of three possibilities: that you are a partner with him, or an employee of his, or volunteering your work for him. The contract determines which of these possibilities applies, and if there is no contract, then custom is the arbitrator. So if custom considers them partners, then you are a partner. If it considers them employees, then you are entitled to an employee's wage. If it considers them volunteers, then you are not entitled to anything. Your father is not strictly obligated to accept you to work with him again, but he must be just among his children in giving, while it is permissible to favor some of them for a legitimate reason. And helping a father in his trade deserves that the child be given and preferred.