One is it sends the message that this is a business, not only to the IRS, but to yourself. You're not just playing around here. You're serious. Once you have a business name, you start thinking that you really need a business plan. And a budget.
That budget thing is important. If this is a business, there's going to be a profit... or a loss. Probably more of the latter to begin with than the former. You need to decide up front how much money you're going to commit to this new business. It's much too easy to spend more than you can afford unless you separate your personal funds from your business funds. So you need a separate bank account. In the name of your publishing company.
There are a few restrictions on the name for your business. You'll have to register it as a DBA (Doing Business As) with the state where you live. It can't be a duplicate of an existing business in your state. You'll also want a domain name, a.k.a. a web site address, that's your business name. This one has to be unique in the world in one format or another.
Do you have any idea how hard it is to come up with an original name that hasn't been duplicated in one of those two areas? With the explosion in self-publishing and small presses, lots of names that you'd normally think of have been taken. Naturally the first things that came to my mind are things uniquely Arizona. Old Pueblo, Saguaro, Prickly Pear. All taken. I thought for a brief period of time that Prickly Pear Publishing would work, but there's one of those in Texas. Petroglyph Press has a certain ring to it. Nope. They're in California.
Another type of name I've been thinking about is a pen name or two. Now, with traditional publishing, it's usually your agent or publishing house who tells you you can't write under your own name. Sometimes this is because the book series is a work-for-hire. The publisher comes up with the idea for the series and hires an author to write it. Since it's their idea, they want to own the author name it's published under. It isn't yours. Other times it's because you're writing for two different publishers or in two different genres.
That last part about different genres has some merit. Readers tend to identify an author with one type of book. If they like an author, they want to buy everything they've written. This is a really good thing. Except when author Jane Doe writes inspirational women's fiction and erotic romance. The inspirational readers will be shocked at having bought an erotic romance. The romance readers will probably hate the inspirational fiction. Both sets of readers will vow to never buy another book by Jane Doe again. So, to make it easier on the reader who's shopping for another book like she read the last time, Jane Doe would be wise to write one genre under a different name.
The same kinds of restrictions on publishing company names also apply to author pen names. You don't want to have the same name as another author. And you want to be able to register your alternate author's domain name.
Names are important to me. Holly Lisle believes that you shouldn't think about naming a character until after you've fleshed her out and know enough about her to choose a name that fits. That doesn't work for me. While I'm deciding on hair and eye colors, body type, occupation, personality traits, etc., I can't get a good idea of what those are until I figure out what her name is.
So I spend hours thinking about who am I and what business I'll be working for. I know that, like the names of my characters, the right names for my publishing company and pen names will eventually come to me. I just wish it weren't such hard work.