He is the son of a lawyer. This is giving the son some recognition as coming from a a family with a lawyer as the parent. He is a son of a lawyer. This is just stating a simple fact, and no special recognition should be inferred from it. The third sentence could be rewritten as: He is a lawyer's son. Here we are using lawyer in the genitive case to describe "son". This again puts emphasis on the fact that a lawyer is in the family.

I know what you mean with the joke but these sentences were given without any other context and I'm sure that they were created just to practise grammar. The only joke here is on me becuase I was trying to figure out the right answer :D Thank you, Dave!

You're right. I'd say it's completely incorrect. At least I can't think how "He is son of a lawyer" would sound correct in any context. At least it sounds very artificial and "epic" to me that way. Vincent's answer is good. Saying somebody is "a son of a lawyer" doesn't sound very natural. A son as opposed to a daughter? I don't know. It's hard to think of why somebody would actually say that right now. Besides that, maybe somebody was making a joke and used the word "lawyer" instead of a more vulgar word in the phrase "a son of a lawyer". Polish has a similar expression, so you probably know what I mean there.

If I understand correctly, the third one is completely incorrect, right? Weird, I was supposed to point out the difference in meaning between them, so I assumed that all of them are in some way correct. Thank you for the answer!

The third sentence omits a needed "the" or "a." Technically, the first two suffice. However, people only say the first one - "the son" - to emphasize that the person is the spawn of a lawyer. The second one could presumably mean that the lawyer has several sons, but the sentence is usually used in the "the son" sense of emphasis on the father's profession.