Take extra care to double-click directly on top of the layer name itself, not in an empty space.
To change the layer name, your double-click must land completely inside the layer name target area in the panel list. If the pointer is slightly outside the layer name (even slightly above or below the name) you'll be clicking out in the larger list area, where double-clicking opens the Layer Style dialog box. My guess is that when double-clicking with the pen, the stylus is slipping slightly outside the layer name and so you open Layer Style instead.
You might also want to experiment with the Tip Double Click Distance in the Wacom tablet preferences.
