For me the pinnacle of human achievement would be to understand how consciousness works, and this is what I would like to see preserved if possible.
We're not there yet, but the progress since 2000 has been so rapid that I believe we should achieve it in the next few decades. The problem I see is the sheer amount of technology required to get there (for example the Big Brain project with it's supercomputers). We need advanced technology to do the modelling of protein structures, simulation of neural networks, and 3-D visualisation, to name just a few examples. The computers we use now can't be designed by hand, they needed the previous generation of computers to build them, and so on back to the 1950's. I suspect that consciousness will turn out to be a quantum phenomenon, so in parallel with neuroscience, the ability to do quantum physics research, with all the technology that implies, will also be necessary.
I suspect that a large population buying consumer technology is needed to generate the wealth to be able to achieve this advanced level of development (the world only seems to be able to sustain a small number of global computer manufacturers). Therefore it's not clear to me how it could be rebooted by a greatly reduced population, even given all the ingenuity Lewis's book describes.
My strategy, for what it's worth, would be to stockpile a large number of Raspberry Pi's, memory cards and screens, so that as one breaks there are spares to replace it. Some could be used as digital encyclopedias, while others were being used to bootstrap the next level of technology. This doesn't totally solve the recovery issue, but it could be enough to preserve knowledge across a few generations while the world economy was being rebuilt, allow programming skills to be maintained, provide a design capacity and components for control systems.
I have to say that I find the idea very scary, that after 4.5 billion years of evolution, our generation could be the one that determines whether this planet can sustain an advanced technology. It hadn't occurred to me that there may not be a second chance, but in hindsight it's obvious - what a responsibility!