Is it possible to set a custom label for the name of a Lightning Component in the App Builder? In the example below, my component is named "DashboardPage" - can I change the label to something else, or do I have to rename the entire component?
Yes. You need to create a design file for the component:
<design:component label="Label For Component"> </design:component>
See the Add a Design Resource to Your Component Bundle section in the dev guide at https://developer.salesforce.com/docs/atlas.en-us.lightning.meta/lightning/components_config_for_app_builder.htm