As you guessed, we'd likely need to make the star fully convective in order to move it to a sort of second main sequence, where hydrogen fusion begins anew at its center; it's imperative that the hydrogen-rich outer layers are allowed to mix with the hydrogen-depleted core. Ideally, this would happen sooner rather than later. Stars which become red giants typically spend up to a couple billion years as subgiants immediately after the main sequence, entering the red giant branch only when their cores become degenerate. Your civilization would want to restart convection before that occurs.
Let's talk about what gives rise to convective zones in stars. There are three mass ranges to consider:
- Stars with masses less than approximately $0.3M_{\odot}$ are fully convective and therefore actually never become red giants. The thorough mixing allows them to exhaust all of their hydrogen by the end of their (extremely long) lives. Their convective zones consist of the entire star because their low temperatures give rise to high opacities, and high opacities are quite conducive to convection.
- Stars that fall into the range $0.3M_{\odot}\lesssim M\lesssim1.3M_{\odot}$ (the outer boundary is uncertain), like the Sun, have radiative cores and convective envelopes - again, low temperatures imply high opacities, but here the cores are too hot, and radiative transport dominates in the inner regions.
- Stars with masses greater than $\approx1.3M_{\odot}$ have the opposite structure: convective cores and radiative envelopes. This is the critical mass at which the CNO cycle, rather than the proton-proton chain, becomes the dominant mechanism for hydrogen fusion. The CNO cycle's rate has a much higher temperature dependence, with $\varepsilon_{\text{CNO}}\propto T^{20}$, while $\varepsilon_{pp}\propto T^4$. Therefore, the CNO cycle implies a much higher temperature gradient than the p-p chain - and a higher temperature gradient is the other major factor that can lead to convection.
Red giants have more complicated structures. Their degenerate cores and shell burning regions are surrounded by an intermediate radiative zone, and finally the enormous convective envelopes they're best known for. Fortunately for us, these convective zones are far from stable. Indeed, at several points in a red giant's evolution, the star experiences what we call dredge-ups, there at convective envelope reaches deeper into the star and actually mixes some of the byproducts of fusion up to the surface. In short, evolved stars may be susceptible to some poking and prodding, and I suspect that the same might be true for subgiants.
Let's look, then, at what I'd guess are our two main options. Helium fusion via the triple-alpha process has an even stronger temperature dependence, at $\varepsilon_{3\alpha}\propto T^{41}$. That might seem conducive to convection, if it could be exploited. On the other hand, that's helium fusion, not hydrogen fusion; as such, it basically defeats the whole purpose of trying to jump-start a second main sequence!
Perhaps we could attempt to raise the opacity in the radiative zones of a subgiant's envelope - whether they be more akin to those of a red giant or of a massive main sequence star. It seems unlikely that we could manage to reduce core temperatures significantly, but perhaps instead would could change our opacity source. Unfortunately, radiative opacities (with the exception of temperature-independent Thompson scattering) tend to decrease with increasing temperature, as noted above, sometimes as severely as $T^{-7/2}$, in the cases of free-free, bound-free and bound-bound absorption. (They do, though, tend to increase with density!) An exception is electron scattering, ordinarily the major source of opacity in the radiative zone.
My proposed solution is to attempt to increase the metal content of the radiative zone. Most radiative opacities show linear or quadratic dependence on the mass fraction of metals in the gas; therefore, raising the fraction of heavier elements, $Z$ would increase the opacity. These notes indicate that in the scenario where electron scattering is the dominant source of opacity, the temperature and density are related to the mean molecular mass $\mu$ by $T\propto\mu$ and $\rho\propto\mu^4$. I believe that these two dependences mean that changes in temperature and density caused by changes in molecular weight would cancel out, and it's just the explicit $Z$ or $Z^2$ dependence that governs opacity changes.
How could we do this? Perhaps crashing several spare (!) terrestrial planets into the star would help; there are cases where a star has exhibited a noticeably higher metal content that appears to come from engulfing a planet. Ideally, if the subgiant's convective zone was deep enough, metals could be mixed downwards to the upper edge of the radiative zone; from there, it's possible that they could enter the zone itself.
This is all fairly handwavy, but it's not outside the realm of possibility. Increasing opacity could, at the very least, make it more favorable for convection to become more important in any radiative regions, and if the civilization gets lucky, perhaps hydrogen could get mixed into the core before it becomes degenerate. I don't know how to calculate the relevant timescales to determine how long this weird reborn star could stay on its second main sequence; I suspect it wouldn't be incredibly long, but it depends. Like I said, maybe they'll get lucky.
