Is there a name for $(x, f(x))$? Given a function $f$, and an $x$ from its domain, is there a name for the pair $(x, f(x))$?
Is there a defined terminology for one (any) such pair? I think the set of all $(x, f(x))$ is called the graph of the function, but I am asking is there's a name for one (or any) point of the graph.
I would call this an element of the graph of $f$, or an element of $\Delta(f)$ where $\Delta$ sends symbols to their denotations, or an element of $f$ as a set.
This amounts to the same thing as calling it an element of $f$, as suggested in the comments, except that it makes explicit the act of sending a function to its corresponding set. This operation is a no-op if you think of a function as just a set of pairs. It is not a no-op if you think of it as an ordered triple with its domain, codomain, and graph as elements, or if you think of a function as some kind of non-set entity.
It's called an ordered pair and it is usually written as $(x,y)$ instead of $(x,f(x))$ because $y=f(x)$. It's called an ordered pair because it's a pair of numbers, and the order matter. For example, $(2,3)$ is different from $(3,2)$.
You may read more about it here
https://en.wikipedia.org/wiki/Ordered_pair
Or watch some videos on this topic here
https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-negative-number-topic/cc-6th-coordinate-plane/v/plot-ordered-pairs
https://www.youtube.com/watch?v=YlT726odQcM