Do not hide number when calling friends? A friend has an iPhone 4 or 5 running iOS 7, and has configured her phone to always hide her number when making calls, as a safety measure.

As a result, I can't tell it's her calling, since all her calls says "Private number".

Is it possible to configure an iPhone to hide its number for all out calls except for such and such person in the phone's list of contacts?
Possibly.

Your mileage may vary, as this is a carrier dependent function.  I was however able to achieve the following using two AT&amp;T phones in the U.S.

First I opened Settings > Phone > Show my Caller ID and turned the switch off.

Next I placed a test call to the second phone, the other phone correctly displayed "no caller id".   (Note: it does NOT matter who the carrier is for the second phone, all the magic relates to what the first phones carrier offers.)

Next I google "AT&amp;T star codes" and located this list.  *82 looked like the best choice.

I then modified the contact for the second phone, adding *82 before the phone number.

Calling that contact then allowed my caller id info to be displayed on the second phone, while still being blocked for others.

It's my understanding that *82 is reasonably standard for the code itself in the U.S.  I have no idea which carriers support it however (other than AT&amp;T where I did test).  I also have no idea what carriers in other countries might use, or if it's supported with a star code.  Consult Google for "(carrier name) star codes" or check with the carrier.