I think "a" is better.

Yes, you're correct. THE is used with nouns that have specific meanings, or with nouns that are one of a kind. A, AN can refer to one of many things in general. Using THE in this sentence implies that the speaker is the only person to have a son who is a doctor. Using A suggests that out of the speaker's many sons, one became a doctor. So the second option is the best. I hope you found this to be helpful.

Yes A doctor .

Actually, either could be correct. In most cases, you would just the I have a son . . . etc. If there were a group of parents and the group was asked, "which of you has a son who became a doctor?" you could use the first example. In other words, if there were several sons from several families to choose from.

Thank your for your help. I really appreciate it.