You should take your phone in. They will fix it if they can and it's cheaper for them than giving you a new.
If you try to fix it yourself they can put the blame on you, so don't touch the hardware.
Your best bet is to just place an ad on Craigslist and in the paper in the lost and found sections. If someone tries to claim the phone, have them identify the picture on the lock screen.
I used to work at an Apple Store Genius Bar. We were not allowed to give out any information from our system ever. If you're located in the US and the phone has a SIM, it probably has AT&T as a carrier. They will not be able to give you any info on the phone either.
Just to provide some insight as far as why Apple can't help you contact the whoever might be listed in their system, remember that a computer or iPhone is a transferrable device. We would frequently have customers bring in a device for repair that was purchased secondhand and the original owner's information was still in our system.
Let's say Tommy buys a MacBook Pro and registers it. Later, Tommy sells his MacBook Pro to Kate because he's broke. Kate purchases the MacBook Pro but does not update the registration. A month later, she leaves the computer in an airport and you find it. You take it to the Apple store and they say, "Oh, here is all of Tommy's info. We'll just call up Tommy and he can come get his MacBook back."
Tommy is not a very scrupulous person, and he fails to mention that he sold the MacBook Pro a month ago. He comes and picks it up and now he has Kate's money and his old computer back. Sweet deal, right? So you can see why this isn't something that Apple would be willing to do.
The Apple employee you talked to was not trying to be obstructive when he wouldn't help you out, but there's a lot of liability things at stake here. They were pretty committed to the idea of protecting customer data, sometimes to the point of idiocy when I worked there, so they're pretty much a dead end as far as returning the phone.
Best Answer
If you want to return the phone to the owner after it has been wiped, then you will need to contact the carrier. Use a small paperclip to remove the SIM tray and access the nano SIM card. On one of the sides you should see a printed number which is the ICC-ID. Enter the number here to determine the carrier if the carrier name is not printed on the card. Call the carrier's customer support number and explain that you found a wiped phone and all you have is the ICC-ID number and the IMEI/MEID number. That should be enough for them to pull up the subscriber information if they are willing to do so. The only other information they might need in order to identify the owner is the IMSI in which case you would have to take the SIM card to a phone repair shop to read the number off of the SIM card. As for whether the carrier will help, I don't know.
Note, the IMEI/MEID is engraved on the back case, near the bottom of the phone.