IMHO the easiest way would be to use string.replace. All code in one line

var obj=JSON.parse(JSON.stringify(obj).replaceAll("\"label\":","\"name\":")
.replaceAll("\"choices\":","\"children\":"));

result

{
  "name": "mylabel",
  "children": [
    {
      "name": "mylabel_11",
      "children": {
        "name": "mylabel_12",
        "children": []
      }
    },
    {
      "name": "mylabel_21",
      "children": {
        "name": "mylabel_22",
        "children": []
      }
    }
  ]
}

You could look to use recursion in the following way, noting that the if statement accounts for the fact that your object's "choices" can be either an array or an object.

function replaceObject(yourObj) {
    if (yourObj.hasOwnProperty("label")) {
        if (Array.isArray(yourObj.choices)) {
            return {
                name: yourObj.label,
                children: yourObj.choices.map((choice) => {
                    return replaceObject(choice);
                }),
            };
        } else {
            return {
                name: yourObj.label,
                children: replaceObject(yourObj.choices),
            };
        }
    }
}

I think you're asking for a way to make it go arbitrarily deep into your object. I think this could work:

function recursiveKeyRename(obj) {
  var new_keys;
  for (key in obj){
    var old_key = key;
    key = key.replace("label","name");
    key = key.replace("choices","children");
    new_keys[key] = obj[old_key]
  };
  if (obj.children && obj.children.choices)
  {
    recursiveKeyRename(obj.children.choices)
  };
};

I haven't really tested that, and that only works if all of you "choices" are objects, not arrays like you (maybe?) implied in your example. It can easily be retooled for whichever use case, though.