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I think you are attaching too much importance to this, insignificant, event.

They probably will gossip a bit. They tell this story as what it is - a funny story to have a laugh about. You can't avoid it. Then they will forget.

If they have been working with you for more than a few days, I can't imagine they could start behaving differently towards you just because of this incident.

The more attention you will be devoting to it, the higher the probability people will add a second episode to the story ("Mom344 has been mortified and behaving super strangely as a result"). Just smile with the people, in one week nobody will remember. The best answer is humor. Tell them you brought up your son very well - he doesn't pee on your house.

I wouldn't worry too much about it. To be honest, I don't think it reflects too badly on you personally, and it says nothing about who you are professionally. As a member of your team, I certainly wouldn't judge you for it. Sure, there are always going to be people that do, but that is out of your control.

Your best course of action would be to go to work and act like nothing ever happened. Don't bring it up yourself, and continue to manage your team the way you always have. If someone else brings it up, acknowledge that it happened but entertain the conversation as little as possible in a polite yet firm way. Continue to be professional, and make it clear that the personal shortcomings of your family are not what you are interested in discussing at work. You should appear as if their mention of the event did not faze you in the slightest. This will make them look and feel foolish for bringing it up and make you look like a professional person that should be respected.

With that being said, you should definitely talk to your son about it!

Don't dwell on it. Admit to feeling embarrassed and move on. Your colleagues will respect you the same or more than before.

Their laughter was likely because the level of embarrassment was trivial enough to make it legitimately funny in the moment. Most colleagues would not laugh at an event that seriously jeopardized your character or authority.

Remember, your son peed on a fence. You didn't. And even if it was you, that wouldn't be the end of the world at most companies. Your colleagues found it funny, because it was funny. Most people with children (of any age) have felt embarassed by them at some point, and chances are at least one of your colleagues has peed in a bush at least once or twice. It's an empathetic scenario.

How do you respond? I like Fattie's answer. Admit your feelings of exasperation at your son and then move on:

"So embarrassing!"

or

"I was mortified."

But here's where I'd like to take a moment and say that, if it were me, I'd welcome small embarassing events like this on occasion. Some of the best leaders I've worked with use moments like this to build rapport. Exposing slight vulnerabilities increases the comfort level between yourself and others. It's humanizing.

If you're a storyteller, you can spin a yarn for colleagues that you feel close enough with to have a laugh. Here's an edit on your story. Imagine yourself in a casual setting with a colleague who was there and one that wasn't, both individuals you're somewhat comfortable with:

"We had an event where the whole company got together - EVERYONE! About 200 people! We had dinner, a live band... the works. My department had rented a small bus, alright? I was the first stop on the way back. And when we pull up to my house, and there's a figure peeing in the shadow. Disconcerting, right? Well, illuminated by the car's headlights, I recognize him. It was my son. Yep. My 27 year old son. Drunk and stumbling about. My color blanched! I mean, what the hell Son344! This was so embarassing! All of the people in the bus started laughing. But here's the icing on the cake: he started calling out to me! He made sure every last person on the bus realized it was my son! It was awful! I was absolutely mortified!"

The point is: you don't dodge trivialities. Either admit embarrassment and move on or own it completely. Recognize that the event was funny because it was trivial. You share a somewhat vulnerable story that couldn't damage your career anyway and gain rapport. In the end, you and your colleagues are human, and the workplace moves on.

Don't worry. 90% of your colleagues fall into two categories: Those who have kids, and those who will have kids. What they have all in common: They are relieved that it was your son and not theirs.

It's really no problem. I can tell you what a real problem is (happened to a friend of mine when he was 17). He came to school one morning, gathered all his friends, and told them "I'd rather tell you now before you read it in the newspaper, but my brother has just been arrested for murder". So you just go to work, and if anyone makes comments, you ask them if they have kids, and whether they have never, ever done anything embarrassing.

As long as the jokes are in good taste and don't cross the line, I'd just roll with it. If they do cross the line, I'd approach the person with a witness and ask them if there is anything they need to discuss with me. Afterwards, I would firmly state that I will not tolerate the humiliation of myself, my family, or fellow colleagues at the workplace and that if it occurs again I will formally escalate the situation through the proper channels.

Never be afraid to stand up for yourself or people you care about, even at work. They aren't very good colleagues if all they can do is poke fun at you and not recognize that it is embarrassing and upsets you.

I don't have children, but in general if someone tries to bring up a subject that's unpleasant for me I try to reflect that by making it unpleasant to keep using that subject in a conversation.

For example, on the softer side I may just listen and refuse to contribute more than necessary (so it doesn't go from unpleasant to awkward), in effect causing the conversation to die naturally, which typically results in either that or the conversation partner changing subjects to keep it alive.

To make it more unpleasant you might also reflect the negative feelings that the subject evokes by presenting an alternate side to you that they won't want to deal with. For example, you might share whatever you feel comfortable in sharing about it while using a harsh tone about it (in the case it makes you mad). That tone shouldn't (and doesn't need to) be directed at them, unless what makes you mad is their reactions. The point is not so much communicating whatever your words technically say, but rather showing that if they want to keep that subject they'll have to deal with the unpleasantness it evokes in you. Nobody wants a negative conversation partner.

Hmm... I may be overgeneralizing on the above paragraph, so I'll tell you: the example of a side I'm thinking of is a strict side. Sharing how you dealt with the problem and ensured that it never happens again in an assertive "this is what's going to happen" tone, might cause them to try to avoid that side of you (in part by changing the subject). Wanting to be strict might not be what you're feeling though, and it might not be part of your personality, so I'm not sure what would be an appropriate tone in that case. It's important to be natural, so I wouldn't try to pretend feelings.

I generally like to be honest this way, instead of faking jokes and whatnot.

Another option you could go for is being blunt and unafraid to bring up awkwardness. I mean, you could flat out stop the subject dead in its tracks on your own instead of trying to get other people to do it. You don't need to pretend that you're ok talking about it if you're not. You talked about respect. People respect those in control, so take control.

The importance of this event is only in your mind. Consider watching this video: https://www.youtube.com/watch?v=Al6avEKEpJ0&t=3s