Attach this script to your gameobject/prefab to enable a simple drag mechanism for your UI elements. You'll also need to add the EventSystem object to your scene.

It's fairly straightforward. The overridden UnityEngine.EventSystems.OnPointerDown and OnPointerUp methods simply toggle the dragging bool. Then the Update method will modify the position of the object this script is attached to.

With this solution, the UI element's center will snap to the cursor, but to make this cleaner, an offset can be added based on the position within the UI element when the OnPointerDown is triggered.

using UnityEngine;
using UnityEngine.EventSystems;
using UnityEngine.UI;

public class UIElementDragger : EventTrigger {

    private bool dragging;

    public void Update() {
        if (dragging) {
            transform.position = new Vector2(Input.mousePosition.x, Input.mousePosition.y);
        }
    }

    public override void OnPointerDown(PointerEventData eventData) {
        dragging = true;
    }

    public override void OnPointerUp(PointerEventData eventData) {
        dragging = false;
    }
}