Joking aside, I don’t know of a process. My main criteria for a name:

1. Short and memorable

2. Easy to spell (but I’ll make some exceptions if it has great marketing potential like faaans did)

3. You can get an idea of what the business/ product is about just from the name.

Domain name availability is nice but not a deal breaker.

The most important factor I think is how well you can recall it in your mind after it’s been a while.

“What was the name of that … oh yeah : {name}”