Hey, that's a great app indeed!!!
For SEO/Traffic etc., here are some more ideas:
1. This is 2016 and I don't know what are you doing without ANY social media presence (at least, none that is visible on the site)? With the kind of service that you have there, LinkedIn and Twitter should work well. Insomuch as social signals is one of the SEO factors today, this would help SEO to some extent as well
2. Consider creating a blog section on the site and publishing relevant posts periodically
3. How about creating a widget/plug-in type functionality using which other sites can copy-paste some code to display a box saying something like: Generate a Great Company/Product/Domain Name, with the basic input features of your app and a Generate button and then, when users hit the button, they reach your site where the results/name suggestions are shown? With something like this, I am reasonably sure you should be able to interest at least some domain registrars and hosting companies to bite!!
HTH. I'll be happy to volunteer further/hands-on promotion/SEO/SMO work in case you're interested - let me know, and all the best.