The poor thing! It can be bothersome, can't it? I'm so sorry if you feel your pediatrician is brushing you off. I know as a new mom, you don't want your child to be uncomfortable and want to be able to make it go away. I would recommend a visit to a pediatric optometrist or ophthalmologist (yes, such doctors exist!), who will be able to evaluate your daughter's eyelids.
OK, I am an optometrist. I haven't evaluated your daughter, so I can't tell you what to do. But I can say that I see this often as well, especially in children with long lashes or from certain populations (I see it in Asian and Hispanics a lot).
There are two possible conditions going on here, both are related. Entropion is a condition where the eyelid turns inward and can cause the eyelashes to turn inward as well and brush against the eye. Trichiasis is a condition where the eyelashes actually grow misdirected and, as well, can grow inward. Anyway, same result in that the lashes brush against the eye. Both conditions are not uncommon at all. Very, very, very rarely are they caused by a medical condition.
Yes, as her eyelids grow, often the condition will outgrow itself and the eyelashes will start directing outward "normally". So, unfortunately, we have to just wait it out, wipe her tears or give her non-medicated artificial tears (absolutely not the red-eye "Visine/Murine/Clear-Eyes" type drops!!) if she seems irritated.
A pediatric optometrist or ophthalmologist can perform a procedure called epilation. It's just like tweezing. It has to be performed very carefully to make sure the lash does not break, which would cause even more discomfort or even make it worse and cause a keratitis (a type of eye infection) when the broken end pokes her eye. I perform this "procedure" to patients all the time, although I don't even think I could do this to my own son. Babies either love it when they realize they feel better, or they hate it.
When she's older and if the condition persists, contact lenses can shield the eye and provide relief. Also when she's older, sometimes surgery can be performed to turn the lid outward. This is called a blepharoplasty.
Anyway, you didn't want to hear doctor-speak so...ugh, unfortunately, most of the time, we just have to wait out this condition. Perhaps you can get a referral to a pediatric optometrist/ophthalmologist in your area just to check things out. I hope this helps.