I've gotten a little further with this. I can easily add an option by creating a new module with this in /app/code/Vender/Module/etc/widget.xml:
<?xml version="1.0" encoding="UTF-8"?>
<widgets xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
xsi:noNamespaceSchemaLocation="urn:magento:module:Magento_Widget:etc/widget.xsd">
<widget id="products_list">
<parameters>
<parameter name="template" xsi:type="select">
<options>
<option name="homepage" value="product/widget/content/homepage.phtml">
<label translate="true">Homepage Template</label>
</option>
</options>
</parameter>
</parameters>
</widget>
</widgets>
But Magento won't use the template unless I put it here: /vendor/magento/module-catalog-widget/view/frontend/templates/product/widget/content/ which doesn't seem right. I would think it'd be better here: /app/code/Vender/Module/view/frontend/templates/product/widget/content/