Lucky's solution can be done with any sort of short, black tubular thing. E.g. an old film canister, or a stub of wood or dowel. Use some cheap wire snips to open up the mesh to accommodate the light-mount's adjustable strap about 1/2 inch below the top-front edge, and wrap the strap around the black object and the basket (through hole and over the edge). Cinch it tight and you'd have as much rattling as you would have from attaching to the handlebars in the first place. You could even use some rubber from a used tube or other bike-reflector/light clamps and put it between the block and the basket.
[Basket below]
=====< [light, on "O" tube]
/--\
| ||O|
[Bike here, | \--/
forward | |
is to the | |
right] | |
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Another idea is to get a small (3-4 inch) piece of flat metal bar, e.g. a metal bracket with holes (pref. aluminum, plastic). At the bottom of your basket, either on side near front or even straight forward, use wire snips to make a little horizontal slit for the bar to go through, but so it can stay flat with the bottom of the basket. Now you lay the bar there with half in, half out, tied to the bottom with black zip ties or a nut and bolt. You could zip tie the light to the bar itself or maybe screw a piece of wood dowel or pipe to the end and attach the light to that.
[Basket below]
| |
[Bike here, | |
forward | | [light, at end of bar, maybe with a tube.]
is to the | | =====<
right] | ______O
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Depending on how you can recharge the light, and on whether you mind reaching in to press the power button, you could even snip a big enough hole in the front of the basket so the light would simply lie flat (adjusted for angle) on the bottom of the basket, inside, with only the lens up against the outside. This could use black zip ties and be very discrete.
[Basket below]
| |
[Bike here, | |
forward | |
is to the | | [light, strapped inside]
right] | =====<
--------