What makes the variable random is that — unlike the kind of variable we see in a quadratic equation, for example — we cannot say what the observed value of the random variable is until we actually carry out the procedure.
A random variable is a variable whose value is determined by the outcome of a random process.
What makes the variable random is that — unlike the kind of variable we see in a quadratic equation, for example — we cannot say what the observed value of the random variable is until we actually carry out the procedure. We might, for example, sample an individual at random and record measurements of characteristics of interest.
Why are distributions of random variables predictable? How do they arise?
The video below shows a physical demonstration of a random process and the resulting distribution.