Notation is notation. I think it's hard to come up with a single guide because a lot of conventions depend on the area.

In the context of optimization, a star refers to the value of the variable that minimizes (or maximizes) a function you're interested in.

You will also see a hat over a variable (https://en.wikipedia.org/wiki/Hat_operator#Estimated_value) sometimes (from statistics) which is an estimate of some quantity from data (and that estimate may be obtained by minimizing a function - solving an optimization problem)

By the way, this is the first result when googling argmin: https://en.wikipedia.org/wiki/Arg_max