They probably won't, but could by using a distributed pay to host service like filecoin https://filecoin.io/

How sites would make money? I'm not really thinking about that too much right now, but I suppose sites could do all the usual things they do with the current web (advertising, accepting payments via credit card gateways, Bitcoin, Patreon, etc). It doesn't have to be an all-or-nothing proposition and it certainly isn't to me. Some functionality will still require a centralized service​, just not the entire thing.

With some sort of "proof of upstream" cryptocurrency that trickles a percentage back to the content's origin node along with other nodes in between. Users would earn credits by leaving their devices connected and forwarding data, paying most of it back into the network when downloading data.

The original uploaders of very popular content would accumulate a surplus of credits, which they then sell to to mega-crawlers. Content creators would do well to be the original uploaders of their own content on such a network.

The credits would also be incentive to buy devices and leave them running in poor-coverage areas. E.g. the owners of billboards could mount repeaters along rural highways.