How can I help a friend find iPhone that is off, using serial number?

The serial number is of no value here.

First, the Find My feature on the phone that is missing must have been enabled.....before.....the phone went missing.

If your friend sees a message of "No Device Found", this indicates that Find My was not enabled on the phone, so there goes the idea of trying to find the location of the phone online. All that they can do in this case is retrace their steps and check at restaurants or shops that may have visited.

If your friend is seeing a message of "Offline", then this indicates that one or more of the following are true:

• The battery has run down
• The phone has been powered off
• The phone does not have an Internet connection where it is currently located

There is nothing that your friend can do about this other than keep checking a few times each day to see if the status of the phone has changed.

You might want to forward this link your friend as well.......If your iPhone or iPad is lost or stolen - Apple Support

The only mechanism Apple provides for you to find a lost or stolen iPhone is Find My. You cannot turn that feature on remotely - it must be enabled at the time your phone is lost or stolen, to use it.

There is no way for you, as an individual, to determine the location of an iPhone from its serial number. In the case of something like a child abduction, the authorities could get a court order to require carriers to help them to track the phone, but carriers don't go to the trouble of that for anything as commonplace as a lost or stolen phone.