What is the name of the set obtained by multiplying a given number by any rational? Given a number, is there a name for the set where each element results of multiplying this number by a rational?
For a given $ n \in \mathbb N $:
$$ \{ r \cdot n \mid  r \in \mathbb Q \} $$

Given any sort of object $x$ for which it makes sense to multiply $x$ by rational numbers, you can call the set in question simply "the set of rational multiples of $x$", or sometimes, "the $\mathbb{Q}$-span of $x$".
However, it should be pointed out that no matter what (non-zero) natural number $n$ you choose, the set of rational multiples of $n$ is exactly the set of rational numbers, i.e. $\{r\cdot n\mid r\in\mathbb{Q}\}=\mathbb{Q}$.
