I have been using this principle for years and in all honestly it has been very difficult to exactly prove its existence. The closest you can get to seeing its possible existence is maybe reading about the Exact Match Domain algorithm. If there was a need to create this algorithm then there has to also be a one name principle and it has to be important....
Here are a few factors that I can name to help you understand the important parts.
1) Repetition of Characters
I have seen with bigger structures like a Chamber of Commerce and my old RelyLocal database, that when there are small fluctuations in the one name of a business, like an added LLC or Inc to the end of a name or changes in the use of characters, that it is not being picked up in search results. Search engines cannot read and only sees a repetition of characters. So, take note of the characters used in your name.
Yext.com is a company that exists to pick up and fix such inconsistencies. So if this was not important and if it did not make a huge difference, this company would not be important. Follow the link and search your company name to see all the listings. With this info, you would need to update manually when considering to change your name.
3) Age of Domain Name
When thinking about changing your business name also keep in consideration the 'age' of your domain. The older a domain becomes the more weight it it has. It is almost like an old bottle of wine. If say your competitor is new and has a newer domain it would take alot of work and time before they would be able to 'compete' with you online. Check your Whois information for free, online, to see how old your domain is.
Google Places is the one listing that literally plugs into Google's brain. This listing, when claimed, becomes your Google Plus business page and is also the info people find when searching Google Maps. To claim this listing, you'll need to first create a google account.
4) Inbound Links
Third Party links can also create havoc on a name change in online searches. So, if you have signed up for an online deal or if someone has written an article or blog about your business in the past 3ish years, then those results will come up in searches. Marketing Grader can check your inbound links under the SEO section and give you an idea of the linking companies. Having an updated blog or something on those sites will help to sway the results.
5) Google Knowledge Graph
Two years ago the above would have been enough, but today we have bigger problems. We recently started talking about Google Knowledge Graph. This is the new box you see on the side bar when searching for something. This info is pulled from all over the internet and, specifically, Wikipedia it seems. You cannot program it manually and need to fix yourself online before your Google Graphs get generated automatically. So, no more slacking or thinking this is not important.
If you need help fixing an inconsistent name or multiple names online, I can help you. Link-building and referral traffic is one of the Marketing Wheels driving your vehicle down the road. This is also, unfortunately, a big job to fix so time and patience is the key. Contact Me when you are ready.