A static website with a dynamic search function? Yes. As alternatives to embeddable scripts from Google or other search engines, you can provide your visitors a custom search by indexing your content files directly.
* [Hugoidx](https://github.com/blevesearch/hugoidx) is an experimental application to create a search index. It's built on top of [Bleve](http://www.blevesearch.com/).
* [GitHub Gist for Hugo Workflow](https://gist.github.com/sebz/efddfc8fdcb6b480f567). This gist contains a simple workflow to create a search index for your static website. It uses a simple Grunt script to index all your content files and [lunr.js](http://lunrjs.com/) to serve the search results.
* [hugo-lunr](https://www.npmjs.com/package/hugo-lunr). A simple way to add site search to your static Hugo site using [lunr.js](http://lunrjs.com/). Hugo-lunr will create an index file of any html and markdown documents in your Hugo project.
* [Algolia](https://www.algolia.com/)'s Search API makes it easy to deliver a great search experience in your apps and websites. Algolia Search provides hosted full-text, numerical, faceted, and geolocalized search.