iTunes could not connect to iPhone

redmelons wrote:

Does iPhone iOS 10 work with OS X 10.6.8 and iTunes 11.4 please?

No. This is your issue.

iOS 10 requires iTunes 12.5.1 or later to communicate to it. iTunes 12.5.1 requires OS X 10.9.5 or later. (The current Mac OS is 10.12 Sierra.).

You should, in the future, keep the Mac up to date on a regular basis.

For right now, upgrade to MacOS Sierra if your hardware can handle it.

How to get macOS Sierra - Apple Support

Then open the App Store and do all updates, and you can use iTunes 12.5.1.

Of course, jumping from OS X 10.6 Snow Leopard, which is now over seven years old, to Sierra is a massive jump. Software may break, other software will have undergone tremendous changes.

UPDATE - I now see you said you have a 2006 Mac Mini. That machine cannot run OS X Sierra. Mac Mini must be mid-2010 or later to run Sierra.

UPDATE 2 - I see you cannot even upgrade a 2006 Mac Mini to OS X 10.8 Mountain Lion, which would be your gateway to attempting to upgrade to OS X 10.9 Mavericks, so you are just plain out of luck. This machine can never communicate to iOS 10 devices.

iOS 10 requires iTunes 12.5.1 or newer which in turn requires OSX 10.9.5 or newer. Software Update only checks for updates for the current system version you are running but that itself may be out of date. It may or may not be possible to upgrade your computer to the required system. Find your computer model on the http://www.everymac.com web site and near the bottom of the system specification section it will say what OS versions it is capable of running. If you cannot run a newer system (which your 2006 probably can't) you will not be able to sync this phone to your present computer, though some features may be available with iCloud services such as Apple Music or Match. OSX 10.9 and 10.10 are no longer available. Check this article on El Capitan (OSX 10.11) and Sierra (OSX 10.12) upgrade options https://discussions.apple.com/docs/DOC-10692 These systems may run slower on older machines and require additional RAM purchase (4 GB is a realistic minimum and many recommend 8GB). Doing a big jump in system versions is also more likely to affect old software.

Snow leopard to el capitan, will it cause my macbook to be slow? - https://discussions.apple.com/thread/7412959

Current OSX general upgrade information, including system requirements - http://www.apple.com/osx/how-to-upgrade/

If you have any PPC generation software that has been running under OSX 10.6 it will not be supported under any later system versions.

Essentially you will either have to buy a newer computer or do without direct syncing and several other features where you must connect to iTunes directly. As mentioned before you can still use iCloud to do some of these such as backups, syncing and sending photos via Photostream.

Have you tried using Image Capture? I know I once was able to download images from an iPhone using an ancient computer that way.

redmelons wrote:

... but it would be a shame to dump a perfectly good computer ...

Yes, I still have my perfectly good, fancy (Keuffel & Esser) slide rule from decades ago.

But technology does progress, more rapidly all the time. You say you haven't backed up your photos ... what else from your iPhone have you not backed up, that would be painful if your iPhone hardware were to fail? What new Mac operating system features, applications, security measures, etc. are you missing by using a ten year old computer? You've gotten your money's worth out of your computer, for sure. Besides, as has been pointed out, when the inevitable time comes to refresh your hardware, you are going to have to make a huge, quantum leap in your various applications.