Let me have another go, because I really love this question. Essentially it's like a SETI@home project - without going into details, let's just consider the theoretical model that some signals arrive from a source that is sufficiently random and then we have to do some computation on it to get some data. (If you prefer groundhogs to E.T., the weather forecast also has this property.) Then whoever computes the data first, will get a coin.
The main issue I see with similar models is that they don't satisfy property 1 (verifiability). I thought this could be solved through some court, but as pointed out by Joseph in the comments to the first version of this answer, this would lead to problems. So instead, we should pick a problem that can be verified. Unfortunately, I don't know much about what SETI was doing with the data, but probably there was something that they were looking for (and now I don't mean aliens, but some signals). So we could say that every 10 minutes there's a data that comes from space, the first person to find a signal wins a coin, but the data might contain no signal, in which case nobody is rewarded - I don't see an issue with this happening sometimes.