If you have no information, you choose a prior that reflects the fact that you have no information. One way of doing this is the principle of maximum entropy, which tells you in particular that if you're trying to choose a prior over a parameter that lies in the interval [0, 1] (the bias of the coin), the maximum entropy prior is the uniform prior. If you have other information, you incorporate that into your choice of prior.

Laplace's rule of succession is for this problem. Laplace defines orthodox bayesianism, not the heretic (infidel?) you quote. Why shouldn't I believe that sometimes objective probabilities make sense, even if not all probabilities are objective? In any event, I can choose it as a model of the coin. I have a prior over biases and update it every time I see a flip of the coin. Ahead of time, my prediction for each flip is the same, but after a few flips, my new prediction is different from my old prediction. Of course, I could have other models, like that Alice always flips heads and Bob always flips tails, but if I'm flipping and I know I'm not cheating, then a constant bias seems like a pretty good model to me.