I think, domain name affect the website success. Because if the website name is clarifies the purpose of your website then visitors get attract towards your website. It should be catchy. It depends on the marketing approach too. If you are promoting your website in unique way then name doesn't matter.
Here are some tips here -
1. Make sure you will choose the domain name related to your site. That is if you would like to create the site related to games then from it's name, end-user should get the purpose of the site.
2. Make sure to choose the domain extension carefully, If you are developing the educational site then the extension should be .edu so it will clarify the use of your website.
3. You can use country specific extensions too.
4. Use the top rated keywords while assigning name to your website so it will help you to rank your website in SEO.
5. Domain name should be shrot and sweet, easy to remember.
Domain name popularity depends on factors like visitors, alexa, design, architecture, contents on the website , images and number of links pointing to your website, social site linking. You will need to review the all links connected to your website. You should remove the bad links that is spam links. Internal linking should be proper.