There is one answer to your question that is classical, discovered by Dirichlet. The number of proper representations of $n$ as a sum of three squares can be expressed as a sum of Jacobi symbols, for example
$$
r_3'(n) = 24\sum_{m \leq n/4}\left(\frac{m}{n}\right)
$$
if $n \equiv 1{\;}(4)$. Here $r_3'(n)$ denotes the number of proper representations, where
$x,y,z$ in $x^2 + y^2 + z^2 = n$ has no common factor. If $n$ is squarefree then $r_3(n) = r_3'(n)$, otherwise $r_3(n)$ is given by a sum
$$
r_3(n) = \sum_{d^2|n}r_3'(n/d^2)
$$
The above formula strongly suggests that there is no simple closed form expression for $r_3(n)$.

Whether this answer really qualifies as nice is uncertain. It is necessary to separate into cases. The formula looks slightly different when $n \equiv 3{\;}(4)$. How it looks when $n$ is even I do not know.

I should mention that Gauss had expressed the number of proper representations of $n$ as a sum of three squares in terms of class numbers of binary quadratic forms. Dirichlet obtained his formulas for $r_3'(n)$ by applying his class number formula.