I might be mistaken - again, I'm not a lawyer.
In the previous link I posted, a family ran a business named Burger King, in active use for several years before the actual Burger King franchise started in a different state and registered at the federal level. They only registered it at the State level, and when the other Burger King franchise spread into their state, the courts ruled that the Burger King franchise, having the federal trademark, had priority over the more local state trademark, and while the original users could continue to use the trademark, they could only do so within 20 mile radius of their original restaurant.
Now, you're operating on the internet... how the internet plays in to local vs state-wide vs nation-wide vs international, I'm not sure.
If your business is registered, then the name of your business has some protection. If you're running a business without actually being registered, and you didn't file a DBA (Doing Business As) like many US states require, I have zero idea what kind of protection the law affords you.
Wikipedia implies you have some protection, but not much:
The law considers a trademark to be a form of property. Proprietary rights in relation to a trademark may be established through actual use in the marketplace, or through registration of the mark with the trademarks office (or "trademarks registry") of a particular jurisdiction. In some jurisdictions, trademark rights can be established through either or both means.
Certain jurisdictions generally do not recognize trademarks rights arising through use. If trademark owners do not hold registrations for their marks in such jurisdictions, the extent to which they will be able to enforce their rights through trademark infringement proceedings will therefore be limited. In cases of dispute, this disparity of rights is often referred to as "first to file" as opposed to "first to use."
The second party would also likely have some rights, if neither party registered. And who knows? He might be able to provide evidence he was using the name before you on some internet forum when he was a kid.
If you really really want the name, register it ASAP (unfortunately costs money). If you are still iffy about the name, try to come up with one more unique and not a commonly used word and not a pair of common words (like Google is unique compared to the English word Googol).