When $R$ is the Eilenberg-MacLane spectrum of a Noetherian ring, thick subcategories are in bijection with specialization-closed subsets of $\mathrm{Spec}\ \pi_0(R)$. Such a thick subcategory is generated by a single compact object iff the specialization-closed subset is actually Zariski-closed (and in that case a generator is given by $H(\pi_0(R)/I)$, for $I$ any ideal corresponding to the closed set). To say that this holds for all thick subcategories imposes a rather strong condition on the ring $\pi_0(R)$; I believe it's actually equivalent to $\mathrm{Spec}\ \pi_0(R)$ having only finitely many points. (Note that although this condition does hold for $p$-local spectra and the corresponding "Spec" has infinitely many points, one for each height, this situation is badly non-Noetherian.)
The same story holds more generally if the graded ring $\pi_*(R)$ is Noetherian and stratifies the category of $R$-modules, in the sense of Benson-Iyengar-Krause. For instance, this automatically holds if $\pi_*(R)$ is a regular ring concentrated in even degrees.