I am not aware of any state in which there is a general rule that an ex-felon may not be able to be the founder or the executive director of a nonprofit corporation.  There may be certain specific situations, such as an ex-felon with a sexual offense conviction who may not be able to work with youth or vulnerable seniors, or a situation where a state license is required but not available to a felon, but there is no general prohibition that I am aware of.  In fact, you often hear of charities in which a prior conviction gives the CEO the credibility and the credential to lead an organization dealing with the situation for which he or she was convicted.  I am not aware of any reason that the situation changes if his father starts the organization.   

If the nonprofit is a charity and will be required to register to solicit charitable contributions, the conviction will have to be disclosed as part of the registration process in most, if not all, states.  Question 14(B) of the Uniform Registration Statement asks if any officer, director, or principal executive has been convicted of a misdemeanor or felony, but the information is for purposes of disclosure and not prohibition on service.  Since very few people ever actually read a registration statement, as a practical matter, disclosure is not likely to cause much of a stir.  But if the organization is proclaiming the conviction as a credential, the registration will be a confirmation of the fact.