Is it useful to consider this slight generalization of the terminal object? Within a given category, an object $A$ is called a pseudo-terminal object if for any object $X$ there exists at most one morphism from $X$ to $A$.
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Of course it could be useful somehow but notice that such an object would not be unique in any way (not up to isomorphism or anything like that). In fact, in an indiscrete category all objects will be pseudo-terminal and no two will be isomorphic. This is some indication that this notion is probably not going to be extremely useful.
It is worth noting though that the notion of a weak terminal object, that is an object $T$ such that for every object $X$ there is an arrow $X\to T$, is sometimes used. In general, weakening 'unique existence' to 'existence, unique up to something' is an important general line of investigation (see, e.g., weakening a topological group to the notion of $A_\infty$ -space). The notion you offer above seems to go in a different direction: Weaken a notion from 'unique existence' to 'unique existence, or maybe no existence at all'. This is somewhat related to the difference between functions and relations (depending on what you have in mind to do with such objects, you might want to have a look at the category $Rel$ of sets and relations).