Since I am being asked the same question repeatedly and since the given answers are not quite correct, I post another answer despite the thread being so old.

According to a talk by Domingo Luna around 1985, the term spherical variety is not derived from spheres, at least not directly. Firstly, spheres are way too atypical, e.g., their compactification theory is pretty pointless. Secondly, in invariant theory circles spheres are called quadrics anyway.

The true origin is a paper by Manfred Krämer from 1979 called "Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen". Krämer had observed that one of the standard constructions for spherical functions generalizes from symmetric spaces to arbitrary homogeneous spaces $G/H$ with $G$ a compact Lie group. If $G/H$ contains not too many spherical functions (i.e., they commute under convolution) then he called $H$ a spherical subgroup and proceeded to classify them for simple $G$ in the paper mentioned above.

Around the same time it was realized (Vinberg-Kimelfeld) that Krämer's condition is equivalent to a Borel subgroup having an open orbit on the complexification of $G/H$, i.e., $G_{\mathbb C}/H_{\mathbb C}$ being spherical. Since Krämer's list does provide lots of non-trivial examples, Brion-Luna-Vust came up with their term.

So the implications are
$$
\text{sphere}\Longrightarrow\text{spherical function}\Longrightarrow\text{spherical variety}
$$
The first arrow this time makes sense since spherical functions were first seriously considered on spheres (Legendre and Gegenbauer polynomials).