I have recently adopted a three month old kitten. One of her eyes is often runny. The discharge has a brownish tint to it and forms a crust on the fur beneath her eye. I have been cleaning her eye a couple times a day to keep up with it. Where we adopted her, the people mentioned that this eye took longer to open than the other. I'm wondering if this is just some kind of birth defect or if it requires a visit to the vet?

I would take her to the vet. You (and I) don't know what is causing the discharge. Better to be safe than sorry (and wondering).