A notion strongly related to this property (if weaker) is decomposability. A decomposable law is a probability distribution that can be represented as the distribution of a sum of two (or more) non-trivial independent random variables. (And an indecomposable law cannot be written that way. The "or more" is definitely irrelevant.) A necessary and sufficient condition for decomposability is that the characteristic function $$\psi(t)=\mathbb{E}[\exp\{itX\}]$$ is the product of two (or more) characteristic functions.
I do not know whether or not the property you consider already has a name in probability theory, maybe linked with infinite divisibility. Which is a much stronger property of $X$, but which includes this property: all infinitely divisible rv's do satisfy this decomposition.
A necessary and sufficient condition for this "primary divisibility" is that the root of the characteristic function $$\psi(t)=\mathbb{E}[\exp\{itX\}]$$ is again a characteristic function.
In the case of distributions with integer support, this is rarely the case since the characteristic function is a polynomial in $\exp\{it\}$. For instance, a Bernoulli random variable is not decomposable.
As pointed out in the Wikipedia page on decomposability, there also exist absolutely continuous distributions that are non-decomposable, like the one with density$$f(x)=\frac{x^2}{\sqrt{2\pi}}\exp\{-x^2/2\}$$
In the event the characteristic function of $X$ is real-valued, Polya's theorem can be used:
Pólya’s theorem. If φ is a real-valued, even, continuous function which satisfies the conditions
φ(0) = 1,
φ is convex on (0,∞),
φ(∞) = 0,
then φ is the characteristic function of an absolutely continuous
symmetric distribution.
Indeed, in this case, $\varphi^{1/2}$ is again real-valued. Therefore, a sufficient condition for $X$ to be primary divisible is that φ is root-convex. But it only applies to symmetric distributions so is of much more limited use than Böchner's theorem for instance.
