When speaking with a boss/manager's son, should I say:
Your father requested so and so.
Mr. Doe requested so and so
I only have a professional relationship with the son. We don't chit-chat.
Interact professionally based on professional roles. Say "Mr. Doe" or any standard professional way of referring to him that is appropriate.
Personal ties outside of work can be a challenge in the workplace for those involved. A parent-child relationship can be particularly difficult. In some cases, the child may struggle to step out of their parent's shadow and be known for their own work, rather than as "the boss's kid".
In such cases, usually the individuals involved are trying to keep the personal and the professional separate. The most helpful thing for you to do is respect the same divide: base your interactions with them on their roles in the company, not on personal factors outside of work. If you say "your father requested this", you might be subtly contributing to a difficult situation.
Note: this may change if you develop more of a relationship with the people involved and interact more casually. But it would be the starting point for professional interaction.
The boss is speaking in his function as the boss and not as the father, so you should rather ignore the relationship.
I run a family business that has employed my children. We are all on a first name basis, so both staff and children would occasionally say "Kate" to each other when referring to me. The kids might say "mum" to someone, no worries. Some of my staff had been with me a long time, their kids were friends with my kids etc, these are the ones more likely to say "your mum" to one of my children. It never once mattered to me a speck.
Note: the largest this company ever got was 11 people. I might have a different answer for a team of hundreds, or when the parent didn't own the company. But for a small cohesive group, where everyone knows the relationships, and many staff have known us for decades (I had a young programmer for a summer job who I first met when he was 4, and have twice employed people who lived close enough to my house to walk to work when the office was attached to the house) it doesn't matter what you call me, everyone knows who I am.
This is a matter of business culture and the personal preference of the people being addressed. How do others refer to the boss to his son or daughter? Have you asked the parent and/or child how they would prefer you to handle the situation. I know there have been people I worked with who did not want to be known as a relative and they preferred the people who knew not to mention it. I have worked other places where the relationship was always mentioned. Most places I have been on a first name basis with both people and referring to people by their first name to a relative seems much more natural than by saying Mr. Jones.
When I worked at my dad's restaurant in Germany as a teenager and young adult, the staff would always refer to him as "the boss" when talking to me, unless they were on first name basis with him, in which case they would sometimes use his first name. I was fine with that, and I don't think he cared, as long as the job got done and communication worked.
In my view it doesn't matter, both are equally appropriate. The arguments presented by dan1111 and FooBar are perfectly valid, but in most cases you can use either.
I used to say to a subordinate, "Your son is doing a good job and it is really good we've got him as a tester". My subordinate never had a problem with this, and others in the office made reference to the relationship too. While saying "Your son" is coming from the opposite perspective, it is referring to the relationship.
It probably depends on country and company (or family) culture, but in my sector (construction) at my place (Catalonia) there are a lot of family business and it's perfectly fine to refer to people by their relation. In fact, sometimes using the most formal designation is ambiguous: if somebody called to some business and asked to talk to some people using the surname, it would be uncommon to be asked back "Which one do you want to talk to? The father or the son(s)?".
Of course that might be different in family business - proud to be family business, and sometimes proud of their several generations long history - than in a big company, where working close to relatives could be seen just as a sign of nepotism.