I am trying to sign out my Apple ID to enable me give away my iPhone 5 but anytime I try to I get a message telling me that; verification Error. The iPhone cannot connect to the server. I have tried several times but still.

Some things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Restore - If you see an error when you update or restore your iPhone, iPad, or iPod

Backup - Restore your device from an iCloud or iTunes backup.      

If you are giving the phone to someone else, so the steps in the article blow.

Selling/Giving Away Device.