Here's a solution I've used on a project to add a user switcher dropdown with typeahead search to a navbar. It uses the pg_search gem for Postgres full-text search.
There are a lot of blog posts out there with similar solutions, with terms like: typeahead, autocomplete, autosuggest, instant search.
class UsersSearchesController < ApplicationController
@users = users_query(params[:query])
respond_to do |format|