The 3 posters above me brought up some really good points. For emergency fund purposes, a piggy-backing card from your existing account would be the best since it probably has a high limit, and even better if it is a Visa/Mastercard. For credit building purposes, credit limits don't matter, so your daughter can go the student card or secured card route. If you have some cash to set aside for a year or so, get her to apply for a Visa/Mastercard secured card with a reasonably high credit limit; a rock killing two birds.
I'm 19 and studying pharmacy, so I guess I can relate to her. I would apply for citi forward or discover it first, If denied, setup a bank account with Wells fargo and there's 99% chance that they will give her a CC with 1000-2000 CL if she has a part time job. Don't apply for any other cards because the chances are they gonna denied her for lack of credit history. I do have to say Citi has bad customer service while Capital1 is stingy with Credit limits. If nothing works, add her as a AU on your credit card so she can start building her credits.
For your daughter's travel abroad, the Bank of America Travel Rewards for Students might be a good choice,
this card proved very dificult for me to get when I tried in the beginning, flat out denial. I just got approved recently (my TU08 was 693 but I don't remember what they pulled) and they only gave me a $700 limit.
Yikes, maybe not a good choice for a first card then.
The card was designed for blank slate thin files. This one was going to be my recommendation too, student card with no forex fee and EMV chip, smells like victory for a student going abroad for a month.
To the OP: pick a couple cards and have her apply for them. I'd probably do BOFA, Citi Forward, and then maybe hold pat for now. More options open up at six months or a year, and 2 cards is plenty to get someone started.
Alternatively if you want to really control the limit, could set her up a secured card (again BOFA travel rewards, or SDFCU's secured card which was tailor made for overseas usage and possibly #3 on the secured cards list as well), but I don't think it's necessary in this instance.
Longer term, her gaining payment history is what's going to help the most for her. Limits are irrelevant other than making certain there's enough for the trip incidental expenses.