S.,
I have been nannying for several families in the past and still do. I love this show in England called "House of Tiny Tearaways". Don't know if you have heard of it or have seen it, but it is basically a psychologist living with a family for a week and helping them with questions and problems they have with their children.
About your son, do you have a bedtime routine? I found it is easier to put kids to bed at this age after a warm bath to relax them and then doing the whole brushing teeth/pajamas, and a story that is not raced through. Some of my favorite moments were putting the kids to bed because we would wrap-up the day or just talk face-to-face, etc. And singing the son marks the end and that you will be leaving or have something. I know this all takes time, but it is important. And if you have a husband or caregiver, then, it is important that your son also has a routine and knows this with them.
Also, it is important to say that it is time to go to bed and that you will be there ... and that it is bed time and you need to stay in your bed. Tell him that if he gets up, then, you will put him back in his bed, etc. He needs to know your clear expectations and what will happen.
On the show, the first few nights, the mothers ended up putting the kids back in bed a ridiculous amount of times (sometimes 40+), but by the third night the kids knew what was going to happen. When you put him back in his bed, do it gently and DO NOT talk to him, lay next top him, and leave. It might sound harsh, but it will be hard to do this the first two nights ... but it works! There will be a lot of crying, but this will give you PEACE with yourself and the situation in the long run! He will eventually stay in his bed at night-time.
Some things to consider are to make sure that toys are generally put away, so your son would not be distracted while he is laying in bed. Maybe you could do this before the bath routine?
Another thing, I am not sure what time he takes a nap, how overtired he is during the day, what time he sleeps in, etc.
Or maybe you work during the day and are only home during the bedtime routine and he just wants to be with you as much as he can.
There are so many factors to consider, but try doing the return to bed routine for a few nights and see what happens. IT really does work! It does have a specific name, but am not sure what it is off the top of my head. Hope this helps.