If she is interested in shooting, first spend $300 and buy her a Ruger or Browning semi-auto .22 and a few thousand rounds of bulk, hollow point ammo.
Over a few months, have her (and you together!) shoot at least 1000 rounds at paper and reactive targets - like soda cans or clay pigeons.
Once she has shot her 1000 rounds, take her to the gun shop and buy her the defensive pistol she likes. As mentioned above, semi-heavy is better. ANOTHER GUN would probably be better for CCW, unless she is going to CCW right away. Let her decide on autoloader or revolver. Take her out and have her shoot her weapon of choice a lot. Get her involved in shooting competitions.
Buy her CCW training (even if she won't carry), a good holster for it (even if she won't carry), and a nice locking case or quick-access safe for the two guns.
Keeping the .22 will help her teach others about shooting.
Of course, in the "real" world, all this may not be possible. People have different priorities and interests. I think she will respect you for offering the $$ investment and time, but she just may not have the time or interest herself. She's got a whole new life in front of her, and a Saturday on the Shooting Range takes away from a whole lot of other social options.
If she is not interested in spending some trigger time with the .22 pistol, consider buying her a "standard" shotgun (26" Rem 870), a case, a combination trigger lock, several boxes of ammo, a few trips to the trap range and a father/daughter defensive shotgun training class. A "shotgun for protection" might be easier for her to accept than a handgun.
Getting her a gun is an awesome fatherly act. But if she is not engaged in the process, a rusting revolver sitting in her sock drawer waiting for a girlfriend's kid to come over and discover does no one any good.
If she has zero interest in guns, maybe you buy her some self-defense classes or meet her for coffee one day with a self-defense expert.
With your guidance, I'm guessing she will become interested in shooting ... but maybe not right away. Put the offer out there. I think she will appreciate the offer to help her learn to defend herself - in what ever manifestation it takes.