how do I generate a drop-down list of functions on scripts attached to a game object?

The solution is simple, if you know where to look - it's called Reflection :)

First, you should specify the access level of the methods you're interested in (public, private, etc)

I will give you an answer on how to get the methods, but I will leave the way you store them up to you (you could put them all in one list, you could create a GameObjectEntry, that has a List of MonoBehaviourEntry, which has a List of the methods themselves, etc)

I will just use a list.

The general idea:

1- Get all the MonoBehaviours on your desired gameObject. 2- from each MB, fetch the methods you're interested in.

 var methods = new List<MethodInfo>();
 var mbs = target.GetComponents<MonoBehaviour>();
 var flags = BindingFlags.Instance | BindingFlags.Public | BindingFlags.NonPublic | BindingFlags.Default; // BindingFlags is located in System.Reflection - modify these to your liking to get the methods you're interested in
 foreach (var mb in mbs)
 {
   methods.AddRange(mb.GetMethods(flags));  
 }
 
 foreach(var m in methods)
    Debug.Log(m.Name);

That's pretty much it really :D

Here's some helper methods you might be interested in (they're extension methods, so put them in a static class):

 public static class Extensions
 {
 public static List<MethodInfo> GetMethods(this MonoBehaviour mb, Type returnType, Type[] paramTypes, BindingFlags flags)
 {
     return mb.GetType().GetMethods(flags)
         .Where(m => m.ReturnType == returnType)
         .Select(m => new { m, Params = m.GetParameters() })
         .Where(x =>
             {
                 return paramTypes == null ? // in case we want no params
                     x.Params.Length == 0 :
                     x.Params.Length == paramTypes.Length &&
                     x.Params.Select(p => p.ParameterType).ToArray().IsEqualTo(paramTypes);
             })
         .Select(x => x.m)
         .ToList();
 }

 public static List<MethodInfo> GetMethods(this GameObject go, Type returnType, Type[] paramTypes, BindingFlags flags)
 {
     var mbs = go.GetComponents<MonoBehaviour>();
     List<MethodInfo> list = new List<MethodInfo>();

     foreach (var mb in mbs) {
         list.AddRange(mb.GetMethods(returnType, paramTypes, flags));
     }

     return list;
 }
 }

An example of the first one:

 using bf = System.Refleciton.BindingFlags; // put this at the top to reduce some typing
 
 var publicFlags = bf.Instance | bf.Public | bf.Default; // Default could suffice...
 var privateFlags = bf.Instance | bf.NonPublic; // NonPublic alone doesn't suffice when you need to get private methods...
 
 var publicVoidMethods = mb.GetMethods(typeof(void), null, publicFlags); // returns all public methods of the signature void methodName();
 
 var privateSpecialMethods = mb.GetMethods(typeof(Vector3), new[] { typeof(int), typeof(GameObject) }, privateFlags);

and so on...

The second one works very similar to the foreach loop ^ - given a gameObject, it would return the specified methods.

 var methods = go.GetMethods(... similar input to mb.GetMethods ...);

Hope that was helpful.

If you want to learn about reflection, there's no one better than Jamie King on youtube :)

EDIT:

Was missing this:

 public static bool IsEqualTo<T>(this IList<T> list, IList<T> other)
 {
     if (list.Count != other.Count) return false;
     for (int i = 0, count = list.Count; i < count; i++) {
         if (!list[i].Equals(other[i])) {
             return false;
         }
     }
     return true;
 }