Maybe personal whitelist/blacklist for domains and authors could improve things. Sort of "Web of trust" but done properly.
Not completely without search engines, but for example, if every website was responsible for maintaining it's own index, we could effectively run our own search engines after initialising "base" trusted website lists. Let's say I'm new to this "new internet", I ask around what are some good websites for information I'm interested in. My friend tells me wikipedia is good for general information, webmd for health queries, stackoverflow for programming questions, and so on. I add wikipedia.org/searchindex, webdm.com/searchindex and stackoverflow.com/searchindex to my personal search engine instance, and every time I search something, these three are queried. This could be improved with local cache, synonyms, etc. As you carry on using it, you expand your "library". Of course it would increase workload of individual resources, but has potential to give feel of that web 1.0 once again.