1) a good sitewide search engine. Google's offer is laughable, and Algolia is too developer-centric (requires pushing the data through API). What I'd want is a single input field where I can put my site's main page URL — and get a working search in a few minutes.
2) subscriptions / monitoring. I want to monitor some event or topic, and I want the updates to be delivered to e.g. my WhatsApp/Telegram/Slack/whatever, with smart filtering, refining etc (in lieu of frantically Googling / redditing / refreshing Twitter feed)
3) context-preserving interactive search, that can ask me questions/ refine results.
4) Timeline search interface for news / events / company history etc. I want to be able to put the name of a person, or company, or TV series, and get a comprehensive timeline view of all things happened there.
I have a lot more ideas, and zero free time :(