The most simple way is to put an animated gif image onto a JButton with 'new JButton(new ImageIcon("myanimated.gif"))'
If your game ist runnig in an active rendering loop (as it is supposed to be) you can just make a custom paint()-method for your menu-item which draws an image over it every time the whole frame gets an repaint(). The menu-item can be every kind of AWT or Swing component (Label, Frame, Button, MenuItem, etc.)
pseudocode:
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| JButton menu_item = new JButton() { paint(Grahpics g) { g.drawImageI(...); g.drawString(...); } }
Frame.add(menu_item);
while(true) { menu_item.repaint(); }
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