I found a solution that might help others.
My iMAC runs Lion, 10.7.5. It has iTunes 12.2.2. My old phone was a 4S running iOS 8.4, and working fine.
I happen to also have an iPad Mini running 8.4.
Planning to buy an iPhone 6s, I backed up my old phone to iTunes on my iMAC.
Bought the phone, and the guy in the AT&T store said he needed to upgrade it to the latest software; if I tried to restore it before doing the upgrade, it would turn into a brick that Apple would have to fix; he wouldn't be able to. So he upgraded to 9.1. When I tried to restore from iTunes, the error message said that synching to 9.1 required a newer iTunes. So I went to the iTunes menu bar and chose System Update and got a message that I had the most recent version. Then called Apple support and spent 58 minutes with a woman who had never heard of this problem (despite the hundreds of posts on this support site) and first tried the steps I had tried, then had me download a newer iTunes even though the download site said I needed OS 10.8.5 or newer. Naturally, it refused to install. The support person then suggested I copy my backup file to an external hard drive and plug it into a computer with a newer OS -- not very practical. It did appear that I would have to upgrade all the way to the current OS to back up the phone... not happening, since Apple's newer systems do not support their own software -- Aperture -- and the alternative, Photos, is unthinkable for a photographer. (Should have bought Lightroom in the first place.)
Then a light went off -- you can back up via the cloud. I called a different Apple support #, one dedicated to iPhones, where the rep knew all about the problem (and says it is supposed to be fixed in the next iOS point release, but he had no date for this.) Following his advice, I upgraded my old cloud account (free, 5 GB) for 99 cents a month 50 GB. (Incidentally, you can't upgrade or read/write iCloud Drive with a 10.7.5 machine, but it is easy to do from the iOS version 8 mobile devices.) I took my old phone, turned on Cloud backup, then hit the Back Up Now button-- took about 15 minutes. . Before restoring my data to the new phone, I had to restore the new phone to factory settings, but since it had no data this was no problem. See https://support.apple.com/en-us/HT203977. After you 'wipe' the phone, you get put into SetUp Assistant, and go through all the steps of that, and then it offers you 'Restore from iCloud Backup.' It shows you the backup you just did, you select it, and it takes about 5 minutes. At the end of that time a lot of the apps show their icons on the face of the phone, but aren't really 'there.' The phone starts downloading them, but as it does so, it for the most part retains the settings I had. That all took close to 30 minutes via WiFi. See https://support.apple.com/en-us/HT204184. and also see https://support.apple.com/en-us/HT201274.
After it is all in place, you check each of the apps; some of them require passwords that didn't get saved somehow; I had a similar experience when replacing my old iPad Mini and restoring from iTunes... maybe because I wasn't using Keychain.
Anyway, this was ultimately fine and I love the new phone. It is unfortunate that the first customer support person was clueless - would have saved me over an hour. But in the end it is certainly worth 99 cents a month. (Final note, my phone had 16 GB of memory, but it was busting at the seams; I expected the back up would require about that much space, but it was only 2.9 GB, and in retrospect I could probably have done this with my free 5 GB. I guess the reason this happens is that the restoration doesn't really restore the actual apps; it just leaves the instructions for configuring them, and gets the apps later.)