I read in a homepage about nets, where the author gives an example (minor modification for context)
Example. Given a topological space $X$ and a point $x\in X$, let $N_x$ denote the directed set of neighborhoods of $x$ - orderded by reverse inclusion. We can construct a net $(x_U)_{U\in N_x}$ by choosing a point $x_U\in U$ for each $U\in N_x$. (Notice that this action requires the Aciom of Choice). [..]
What I do not understand is, why the Axiom of Choice is needed (isn't it superfluous?). For every $U\in N_x$, you can "choose" a point $x_U\in U$. This is possible, simply because that each set in $N_x$ contains a least one element (including $x$ itself), i.e. every set in $N_x$ is non-empty. So, the existence of a such point is justified. If I am wrong, please let me know.
The statement "If $X$ is a family of sets and each member of $X$ is non-empty, then we can choose an element from each member of $X$" is the statement of the Axiom of Choice.
Your suggestion that we always choose $x$ is fine, but it is somewhat unhelpful if you try to understand nets, since it's a constant net. Some definitions in topology will talk about sequences or nets which do not use their intended limit.
For example, $x$ is isolated if $\{x\}$ is open, but if it is not isolated, then there is a net $(x_i)_{i\in I}$ such that $x_i\neq x$ for all $i\in I$ and $\lim x_i=x$. But for this we need the axiom of choice.