why would my doctor want me to gain 10 lbs?

Ya I'm 5'4" and 111lbs and still very slender.

According to the BMI formula, you are just slightly underweight. The additional 10 pounds will move you into the bottom end of the normal range. If you think that is too much, and the doctor probably suggested it as a round number, you could put on just 3 pounds and move as far into the normal range as you are currently below it.

Without seeing a picture, I can't say if I would consider you too thin. As long as you have a bit of shape to your legs and arms as well as curves on the body, I would say you are good looking (my computer wanted to leave out an o). Go ahead with the 3 pounds, see how you feel about your appearance and then see if your doctor is satisfied.

I'm pretty sure a doctor knows ways more than a nineteen year old girl. So you should gain those 10 pounds.

Yep , the extra weight will do you good. It will make you less susceptible to diseases and illnesses. After all the doctor knows best :)