The empty set is an $n$-ary relation for every $n$, right?
How should we call a pair $(n;r)$ consisting of some number $n$ and an $n$-ary relation $r$?
To specify $n$ is necessary only when $r$ is empty, but because there are no reason for $r$ not to be empty, I need to specify $n$ explicitly.
Any term describing this situation?
I don't believe that there is any common terminology for what you are asking about, but you should be careful.
Using the common recursive/inductive definition of $n$-tuples, where $$\langle a_0 , \ldots , a_{n-1} , a_n \rangle = \langle \langle a_0 , \ldots , a_{n-1} \rangle , a_n \rangle,$$ it follows that an $n$-tuple is also an $m$-tuple for all $m \leq n$. Therefore an $n$-ary relation is techincally also an $m$-ary relation for all $m \leq n$. (Differences may arise when talking about $n$-ary relations on a specific set $X$, but even here there might be ambiguity.)
Added due to comments below: Ignore the second paragraph.