My teenage daughter has had a swollen left upper eyelid for over a month. Slight puffiness below the eye also. No other symptoms.?

Puffy. This could be a longstanding chalazion, an inflammation of one or more oil glands in the eyelid. Try some warm compresses for a few days and see if the puffiness declines. If not, call your local eye doctor for an appointment.
Time for visit. The persistence of swelling in this area for a month is likely due to chronic irritation of the tear glands. If you've tried warm compresses for a while & this persists it may be time for you to see an eye doctor. Occasionally this can persist for months & only resolve if the glands are opened from the inside of the eyelid & shelled out.

Related Questions

The gland in my left upper eyelid is swollen and I have pain in eye and slight headache?

Evaluation soon. The gland is possibly the lacrimal (tear) gland. Other glands can swelling and tumors can metastasize and grown in the lid. But infection is also possible. You should see an ophthalmologist very soon and get this diagnosed. Read more...
I agree . In the meantime, use a warm compress over the area for 10 minutes four times a day. Read more...