A demo Umbraco v6+ UmbracoApiController (an Umbraco Context wrapped WebApiController). Each request returns an array of strings which would typically be read in by a JavaScript library as JSON.

JsonDropDownListApiController.cs

using System;
using System.Collections.Generic;
using Umbraco.Core.Models;
using Umbraco.Web.Mvc;
using Umbraco.Web.WebApi;
 
namespace UmbracoDemo.Controllers
{
    [PluginController("JsonDropDownList")]
    public class JsonDropDownListApiController : UmbracoApiController
    {
        // /umbraco/
        public IEnumerable<string> GetData()
        {
            return new[] { "Table", "Chair", "Desk", "Computer", "Beer fridge" };
        }
 
        public IEnumerable<string> GetDataFromEnum()
        {
            return Enum.GetNames(typeof(ProjectStates));
        }
 
        public IEnumerable<string> GetDataFromUmbraco()
        {
            //Change ID to whatever node has the child nodes you wish to select.
            //In a production site this would be a parameter of the function
            var rootId = 1054;
            var children = this.Services.ContentService.GetChildren(rootId);
 
            foreach (IContent child in children)
            {
                yield return child.Name;
            }
        }
 
    }
 
    //Included in the same file for readability
    public enum ProjectStates
    {
        Planning,
        Development,
        InternalQA,
        UAT,
        Complete
    }
}