Someone suggested that search.wikimedia.org, which is an API endpoint, should do something better than give an error at the top level—offer a search box, roll over to the Wikipedia portal—something useful.
I'd suggest to make search.wikimedia.org a search engine that searches Wikimedia projects with tabs for encyclopedia, dictionary, images, videos (separate tabs though both from commons), library, news, books, travel destinations, species, etc. and an option to choose language. It could look similar to the standard search engine - a text box with tabs at the bottom and a button to the right.
Various visualization options could be tried such as showing the location of the results on a map or a timeline
It may improve discoverability of sister projects as not all of them are shown in the 'sister projects results' pane on-wiki.