My iPhone doesn't have the option to turn on call waiting, like it should. iPhone 5s

Not all carriers provide the option for call waiting. If you find the setting in your Settings app, then you should be able to activate it. If not, you can try a reset, holding the sleep/wake and home buttons together until you see the Apple logo and then release. The phone will reboot. Check settings then. If you still cannot activate it, then contact your carrier and ask if they support call waiting and if your account provides that service.

Yes, but restore as new is the last resort. Restore from a backup first.

The simple reset that ChrisJ4203 suggests in the initial response above does not lose any data (or "info and personalization").

If your carrier is a CDMA carrier such as Sprint or Verizon, you will not see that option on the phone. No amount of resetting or restoring the phone will change that. You will need to use whatever procedure your carrier offers to turn the feature on or off. With Verizon, it's on by default.