How can I assign a name to a task in TPL

You could relate any object with any object. Here is an extension for Task. It uses a WeakReference so the task can still be garbage collected when all references are out of scope.

Usage:

var myTask = new Task(...
myTask.Tag("The name here");
var nameOfTask = (string)myTask.Tag();

Extension class:

public static class TaskExtensions
{
    private static readonly Dictionary<WeakReference<Task>, object> TaskNames = new Dictionary<WeakReference<Task>, object>(); 

    public static void Tag(this Task pTask, object pTag)
    {
        if (pTask == null) return;
        var weakReference = ContainsTask(pTask);
        if (weakReference == null)
        {
            weakReference = new WeakReference<Task>(pTask);
        }
        TaskNames[weakReference] = pTag;
    }

    public static object Tag(this Task pTask)
    {
        var weakReference = ContainsTask(pTask);
        if (weakReference == null) return null;
        return TaskNames[weakReference];
    }

    private static WeakReference<Task> ContainsTask(Task pTask)
    {
        foreach (var kvp in TaskNames.ToList())
        {
            var weakReference = kvp.Key;

            Task taskFromReference;
            if (!weakReference.TryGetTarget(out taskFromReference))
            {
                TaskNames.Remove(weakReference); //Keep the dictionary clean.
                continue;
            }

            if (pTask == taskFromReference)
            {
                return weakReference;
            }
        }
        return null;
    }
}

You can't really name a Task, but you can name the method which is executed by a Task, which is then shown in the Parallel Tasks windows. So, if naming the Tasks is important for you, don't use lambdas, use normal named methods.

Surprisingly, this works even with Parallel, even though there the Task isn't executing your method directly. I think this is because Parallel Tasks somehow knows about Tasks from Parallel and handles them differently.

You can't name tasks.

The task library is internally using a thread pool, so the threads can't be named. Also your inheritance approach won't work, because methods like ".ContinueWith()" will always create a new task, which won't inherit from your class.