Something like Chaum's blind-signature based ecash, or GNU Taler would be ideal in terms of efficiency, but it's still centralized in distribution. Freenet currently uses something like this.

Another option I was thinking of would be a pretty inflationary (or demurrage) cryptocurrency in which you have some sort of RandomX or other CPU-bound PoW. A web server could act as a mining pool and use mining shares interchangibly with micropayments.

You could do this mining-share method with Monero right now, it's just that you have higher transaction size in Monero and no real analogue to Bitcoin's LN-based microtransactions. Also you would want the cryptocurrency to be more inflationary (or demurrage-based) to promote usage.

Monero's FCMP++ lays some groundwork for payment channels, but it still lacks the nessisary timelocks. Also there was DLSAG which could have enabled payment channels I think, but it's no longer relevant. I also insist that you would need to change the tokenomics to favor greater inflation (maybe you could make coinbase scale linearly with hashrate?), otherwise the miner reward would be economicially insufficient.