Creating (or renaming) a project called "ROOT" allowed me to do this within Eclipse.
When you initially "Run on Server" it will load the app using the long-form URL:
http://localhost:8080/ROOT
But if you navigate to:
http://localhost:8080/
You'll see it hits the same page.
One caveat is that in the Tomcat settings page in Eclipse, you'll need to ensure the Server Location is set to "Use workspace metadata". I initially had it set to "Use Tomcat installation" and calling the default http://localhost:8080/
URL would just load Tomcat's default page (presumably because a ROOT application already exists / takes precedence).
Update
I've found you can achieve the same result without changing your project name. Simply edit your Eclipse project's .settings/org.eclipse.wst.common.component
file and change the context-root property to "ROOT" like so:
<?xml version="1.0" encoding="UTF-8"?><project-modules id="moduleCoreId" project-version="1.5.0">
<wb-module deploy-name="YourProjectName">
<wb-resource deploy-path="/" source-path="/WebContent" tag="defaultRootSource"/>
<wb-resource deploy-path="/WEB-INF/classes" source-path="/src"/>
<property name="java-output-path" value="/YourProjectName/build/classes"/>
<property name="context-root" value="ROOT"/>
</wb-module>
</project-modules>
Where "YourProjectName" is the name of your project, obviously. :)
Second Update
According to this answer on a related question you can actually change the content-root from the GUI in Eclipse but going to "Web Project Settings" in your project settings.