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][1].  For instance, this automatically holds if $\pi_*(R)$ is a regular ring concentrated in even degrees.


  [1]: http://arxiv.org/abs/0910.0642