How would you keep the index up to date? How quickly would news stories appear in it and stale links disappear? Are you going to read the entire internet every day to keep it current?
the problem is this would be completely useless for current events. but querying sites you use regularly, this could be an option. I often end a google search with modifiers like "wikipedia" or "reddit" or "stackoverflow". If I could store indexes to those sites offline that might be useful.
The creator of this offline search could potentially make money by indexing shopping sites like Amazon, eBay, etc with affiliate links.