iTunes (iMac) doesn't recognize iPhone3g

Thank you for the links, but they didn't help. I actually made things worse, by going into iPhone settings and selecting "Reset All Settings" and "Reset Network Settings", not in that order. Now the iPhone doesn't even show up as a device on iTunes, and no amount of restarting and cable connection can reintroduce it. As you said, the issue is in the phone, which is why I wanted to restore it, but can't. So, in addition to the very poor wireless performance, no usb connectivity and no network service, It is very long overdo for replacement. The AT&T bill is paid up and active. Very frustrating and unproductive. Phone was never dropped or misused...

OK, I restored WiFi with modem password, and iTune lists the iPhone as a device, but still displays the message about connecting with USB to back up and restore software. The problem, to reiterate, is that iTune sees the USB connection, enough to know what the device is, but is blind to it with respect to allowing restore operation. Another apparent issue is iS0 6.1.6 on the iPhone 3G, that is somehow causing the "no service" status.

I give up. I can't restore the phone because iTunes doesn't see the USB connection. It sees the iPhone through WiFi, but you can't or wouldn't want to restore the phone that way. Even if I could restore the phone, the latest software is having "no service" issues on older phones. Than I would have to go through a procedure, akin to "Jailbreak," just to install an earlier version...ahh... You win Apple! Your planned obsolescence has, again, won the day and I must buy an iPhone 6 plus. I'm sure, I will love it, until it gets a few years old, and it is put out to pasture; unsupported.

It was the cable. I have 4 or 5 that all work flawlessly for charging, but only one that permits data transfer; explaining why iTunes did not "see" my iPhone for the purpose of save and restore. I am getting a new phone, but I needed to retrieve personally priceless photos and contacts off the phone.

You were absolutely correct. I did have to try a few cables to find one that did more than just charge. But, the 3G had seen better days, and I ended up buying an android Moto E2. Seriously, the ultimate bang for the buck, bargain phone. But now I have the laborious task of finding third party software to allow some interaction with Mac. I immediately ran into a problem when I wanted to add a downloaded mp3 ringtone. Apparently, the popular AndroidConnect (to mac) software that works well transferring, to and from, all recent OSX platforms, except Mavericks, which of course, is what I'm using. Do I really want to upgrade to Yosemite? Have they addressed the WiFi issues, but that is for another discussion. Thanks for the 3G advice.