Methods to quantify randomness (or complexity) in a signal
There are two different concepts:

If you think as your signal as a single random variable $X$ that is emitting values, then what you want is to calculate the Entropy of the random variable

http://en.wikipedia.org/wiki/Entropy_estimation

If you are considering the entire random signal or stochastic process, then you have to estimate the autocorrelation function. The most random signal possible is White noise, in which every sample of the signal is not correlated with any other except itself.

http://en.wikipedia.org/wiki/Autocorrelation