For me the 2 priorities when choosing a business name are:
1. It is memorable, which usually means something unusual or humorous; and
2. It suggests what the business is about.
Having the name available at Companies House is not that important. OK, it is convenient if your exact chosen name is available to register, but if not, you can usually get very close to it by adding a '(UK)' for example. Failing that, just register at Companies House under any name and use your chosen business name as a 'trading style'.
Same with the domain name. You can usually get close enough to what you want by using different TLDs (.co.uk, .com, .net, .uk.com) and/or the use of a hyphen or 2 (but no more). Another approach is to pick 2 really key keywords and use those as the domain name, for SEO benefits. Customers will still be able to find you, as in most cases your own business name goes straight to the top of the Google listings with little effort.
1. It is memorable, which usually means something unusual or humorous; and
I agree!
For me the most important factors are something eye-catching and memorable, and also something that isn't too ambiguous in the way it is spelt. I know 'mis-spellings' are kind of the thing these days, but it's not good if people are typing trying to find your business and can't because it is spelt in such an obscure way!
Note where it says that it ranks 4th..... that was BEFORE it was claimed by the sandbox - now it's nowhere and I have no time on my hands to go getting it out When it finally re-appears from the sandbox it should bounce straight back up.