Since the sending of the money doesn't need to be instantaneous, how about a captcha system kinda like this:
You collect a bank of images, things like group photos, photos of animals, landmarks etc (perhaps simply by scraping the results of image searchs, taking note of the keywords used, and if the site the image comes from got tags for the image, take note of the tags too, ) .
Each time someone tries to request coins from the faucet, the server randomly picks one of the pictures, distorts it a bit, in a unique way each time, taking note of the way it's distorted to remap distorted corrdinates back to original; and then asks the person to click where in the picture somthing is (selecting what based on the keywords used to find the picture and associated tags if any; things like the front left headlight of the car, the nose of the cat, the tip of the Eiffel tower, the biggest digit in the picture, the face of the youngest person in the picture etc).
Remaping the clicks to the undistorted coordinates, the server then starts ranking the clicks in terms of how close to the average of the other clicks for that same feature. But each time a same person refreshes or asks to try a different picture the weighting of their vote for the coordinate of the picture average is reduced, so if someone tries several times in order to find the same picture and being asked for the same feature, to try to distort the average in their favour, their attempt of tampering will not easilly outvote legit clicks.
Perhaps besides the geometric distortion, it might be good to also add other interferences, like hue, saturation and brightness noise and gradients, invert the color in small randomly shaped part, "burn" out little holes here and there, add a few blur gradients (kinda like randrops on the camera lens), overlay small randomly shaped patches with the colors "solarized", the resolution reduced (pixelazing the image) etc; basicly all sorts of image "filters" that would confuse computer vision, specially when combined, but (at least most of the time) still leave the contents of the image easilly identifyable by humans.