I may be missing something here, but the way Voldemort is acting doesn't make sense to me. He's demonstrated that he thinks there's things Harry knows that he would like to learn. It also seems reasonable that he'd expect Harry to continue to learn useful things that Harry could teach Voldemort. So is giving Harry 60 seconds to do this really the most use he could get out of Harry? Indeed Voldemort might know this and be bluffing.

Playing this card may buy Harry time, maybe a lot of time. With this time, I see a variety of possible outs:

• Hopefully the muggles will develop FAI one day, which will have the power to finally destroy Voldemort. To increase the likelihood of this happening, I'd think that Harry a) has to communicate to the muggle scientists to keep this secret so Voldemort doesn't find out about it and stop it. And b) could try to distract Voldemort and lead him towards doubting the usefulness of muggle science. b) unfortunately diminishes the usefulness that Harry provides which may end up in Voldemort deciding to kill him, but this may not be a horrible outcome to Harry given the circumstances and his utility function.
• Make people more rational. It seems that Voldemort enjoys harming people because their stupidity frustrates him. There isn't anyone in the world who Voldemort thinks is actually smart, and thus there isn't someone he actually respects. Harry has been the closest thing to someone who matches his intelligence, and thus he has demonstrated some respect (in a way) for Harry, and has demonstrated deriving some happiness from mentoring Harry. Additionally, in a more general sense, he seems to take pleasure in making stupid people smarter. Perhaps Harry could accelerate the process of making society smarter. Maybe he could use a combination of magic and drugs to make people smarter. (as an alternative, maybe Harry could use the 60 seconds to propose that Voldemort tries making people smarter and seeing if that makes him happy)
• Continue to contemplate how magic really works. There seems to be incredibly potential power there, far beyond what Voldemort's current powers are. Harry already figured out partial transfiguration and how to kill death eaters at the age of 11(?) and within a year of figuring out that magic exists. Perhaps he'd continue to make discoveries like this and might eventually gain enough power to fight Voldemort. This very well may even happen quickly after one insight. Alternatively, maybe he could figure out the secrets of happiness and propose that Voldemort spends eternity in a magical version of wireheading (Voldie's terminal goal does seem to be happiness, not power or chaos). Granted, Voldemort probably is smart enough to monitor Harry to make sure there isn't anything useful that Harry knows that he doesn't. I'm not sure how to get around this. Making him think you're stupid would make him less likely to monitor you closely, but that also would make him not want to keep you around.