This may be a lot of thing you already tried, you need to explain what you discussed ad-nauseum otherwise we cannot work out where to start.
Is the Mac portable? Can you try another internet connection?
It sounds to me like the connection is not finding the server, that could be due to DNS or proxying on the connection. Trying another internet connection could help rule that out.
You can also specify a DNS server in System Preferences to ensure that the one provided by the router or ISP is not the reason for the error. Google DNS is 8.8.8.8, enter that in the network connection settings & retest. Make a note of any existing values that are not grey (grey = supplied by the router).
It could also be due to certificates, have you changed the CRL or OCSP settings (many people freaked out with heartbleed & chose bad settings within Keychain Access's preferences).
The other thing to do is check logs as the process is failing to see if anything appears with clues. Note down the time you start the update to make searching the logs easier.
Use /Applications/Utilities/Console to view the logs.
Check the 'All messages' section, the ~/Library/Logs section for 'iPhone updater.log'
System.log may have other system wide errors if the OS is failing, but you need to be an admin to view it.
Make sure you clear out any personal info from log output. Post extracts here, just try to select the errors & a few messages around them.
