BrandBucket is described as 'The original marketplace for business names and creative domain names. A #lean branding resource for startups needing a name and ventures looking to rebrand'. There are more than 10 alternatives to BrandBucket, not only websites but also apps for SaaS and Self-Hosted solutions. The best alternative is OYZTA. It's not free, so if you're looking for a free alternative, you could try Brandaisy or Namesnack. Other great sites and apps similar to BrandBucket are Namewink (Free), Namy (Free), indiebrands.io (Paid) and Brandroot (Paid).
Other domain generators exist, but they all suck. Namy has much better AI, and is very straightforward. No ads, no hidden fees. Completely free, and the best recommendation engine on the market (and with tons of improvements planned!). Give it a try and see if you like it.
Avoid hours of brainstorming, research while looking for a name. Focus on building your product instead. Indiebrands offers ready-made brand packages so you can find the perfect name for your business in minutes.