(If we restrict to cardinal numbers of well-orderable sets then that of α is the smallest that is not not less than or equal to that of X.)
The map taking X to α is sometimes called Hartogs's function.
The existence of the Hartogs number was proved by Friedrich Hartogs in 1915, using Zermelo set theory alone (that is, without using the axiom of choice, or the later-introduced Replacement schema of Zermelo-Fraenkel set theory).
Hartogs's theorem states that for any set X, there exists an ordinal α such that
As ordinals are well-ordered, this immediately implies the existence of a Hartogs number for any set X.
In 1915, Hartogs could use neither von Neumann-ordinals nor the replacement axiom, and so his result is one of Zermelo set theory and looks rather different from the modern exposition above.
However, the main purpose of his contribution was to show that trichotomy for cardinal numbers implies the (then 11 year old) well-ordering theorem (and, hence, the axiom of choice).