Rigorous definition of randomness
Guess what? There is no such thing! We postulate a space of possibilities and a subset of events and assign probabilities to them. Some people will say that a selection in accordance with those probabilities is a "random" selection, others will insist that only "equiprobable events" can provide random selection.
Actually, mathematicians don't discuss randomness very much. Probability measures, yes, randomness, no.
A rigorous definition of randomness
I think that the only problem with PerfectHacker's definition is his use of the word algorithm.
Otherwise I think it is a good starting point to work form. It seems to me that some definition of the notion of onservation or measurment is necessary -consider quantum mechanics, where a physical process is completley non-random or deterministic (unitary evolution), until somebody decides to measure an observable of the system.