There are two important things about choosing a name for your business. One is legality. The other is marketing.
Legally you need to pick a name that no one else has, then do appropriate research to confirm that the name is unique. Marketing-wise, you need to pick a name that people will remember and have positive associations with from the moment they hear it.
Trademarks, States & Names
Its the trademark issue that triggers the real legal problems. If someone holds a trademark (complete with a logo that is part of the trademark), then no one else can use that name. What's more common is infringing on a registered name, and even then serious legal problems with names are uncommon - basically you just need to check around to confirm that you're not using someone else's name. Even a quick search on the web is frequently enough. Also, depending on the name you choose, bear in mind that few independent, small-business owner handymen are going to hire an expensive lawyer unless you're actually hurting their business.
So long as the two name "owners" aren't in the same state, you're probably OK with being the second John the Handyman in the US.
Here's the deal: If someone else in your state has thought of the same name you have, and has registered that name, they can sue you if you use their business name as your business name. Don't expect to be sued, but don't be surprised when you get certified mail with an official-looking "cease and desist" letter.
Check these sites to see if another company has registered the name you want. Then, when you're sure you're in the clear, seriously consider registering your company's name with one of these services.
1. Thomas Register www.thomasregister.com
2. Niten Research Corporation www.niten.com
3. Mahtta Trademark Company www.mahtta.com
4. Legalname.com www.legalname.com
5. Mark Monitor www.markmonitor.com
Catchy Names 101
Don't name your company anything that a) they can't spell b) they don't understand immediately or c) they have bad associations with. For example, you could name your handyman business "Haephestus's Wrench" after the Greek god of engineering, but most people can't spell Haephestus, and most people will think "what the &^%* is that?" when they see your sign. (Not that they're dumb, mind you, they're just focused on more practical things). It will not be
immediately obvious to people that you're a handyman, unless you happen to live in a town densely populated with Latin scholars. Naming your company after some other obscure Greek god who, say, broke things, would break the third principle of avoiding any bad associations with the name. Why name your company after someone who broke stuff?
Even "Mike the Handyman" is better than "Haephestus's Wrench". It may not be as creative, but everyone will immediately understand what you do. And hey, if you're named Mike, they'll remember your name as well. Other good candidates would be Your Town's Name Handyman, Your Last Name Handyman Services, or Handyman for Hire. One large handyman firm has gotten some press for the somewhat edgy "Husband for Hire" name. That certainly gets people's attention, but it courts the edge of breaking the "no negative associations" rule mentioned above.