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Is there a way to add custom fields to the "Organization" object?
I need to add essentially add a few values that are 'singletons'. I could create a custom object and use validation to ensure it was a singleton.
But is there a better way?

Not yet. Today your only option is a singleton custom object.
I googled "salesforce add organization fields" and this is still the top result. In 2018, you are better-off using Custom Settings (https://help.salesforce.com/articleView?id=cs_about.htm&type=5) or Custom Metadata Types (https://help.salesforce.com/articleView?id=custommetadatatypes_overview.htm&type=5) to handle this requirement. Still, I'd rather be able to add Custom Fields to the Organization object.