Word to describe a mathematical variable that repeats, like an angle or time

What you've described is referred to as "wrapping" and a variable that wraps around is called a wrapped variable.

As jimreed rightly points out, the cause of wrapping is due to modular arithmetic and it is very uncommon to explicitly mention that a variable is "wrapped" because it is assumed to be understood. The only time I've seen mathematical/scientific articles use the term wrapped variable is when they also talk about "unwrapping" the variable. So in your clock example, you would add 12 after each time it completes a full circle to get an unwrapped variable.

I don't have a wiki link to the definition (because, as I said, it's not used commonly). However, the article on wrapped distribution uses the term wrapped variable.

You could use periodic if the repetition follows any kind of function.