I assume that the "offer of the week" from LIDL/ALDI varies from country to country, but last week the LIDL here had pencil engravers available for €9.99. I've only tried this particular model on a beer bottle, but it seemed to get the job done.
An easy way of etching small areas is to paint the surface with a fairly thick layer of nail polish (I remember some fairly strange looks from my S.O. as she was rooting around in my shop one day, and found a bottle of nail polish), and let it dry well. Use a scribe (or similar) to remove the nail polish in the area you want etched (e.g. your name). Take an old phone charger and cut the plug off to separate the leads. Use an alligator clip on one lead to attach to the tool, and a clip on the other holding a cotton bud soaked with a dielectric fluid. (I've used water with table salt, and water with baking soda, both work fine.) Simply rub and press the cotton bud over the area that you've marked with the scribe. If it starts fizzing, you've got things hooked up properly. If not, change the leads. This method, as you know better than I, will produce a bit of gas that you probably don't want to be breathing, so a properly ventilated work area is a good idea.