CM,
If you are you handy with a sewing machine you have a couple great options without drilling into the Flea.
The Flea has a small hole for a hook built into the body. It is located under the neck where it joins the body. You can rig up a neck strap using webbing, string, etc. and hook it in through there with hooks, knots, etc.
Another options is to use industrial strength adhesive velcro on the body side instead of a button. Industrial strength velro can be pruchased at hardware stores such as Home Depot.
On the back of each side of the velcro is a strong adhesive. You peel off the backing and stick in on.
If you put a piece on the bottom of the Flea, you could use webbing to make a strap. Sew on a piece of velcro to the end of the strap and stick it to the velcro on the bottom of the Flea.
Attach the other end of the strap to the neck/headstock by stitching a couple lengths of chord to the neck end of the strap. Then you just tie the chord around the neck or through the headstock.
You can purchase slides to make the webbing adjustable.
The only thing you loose with this second approach is being able to stand the Flea upright.
Best of luck.
