Is eyelid ptosis a weak muscle problem and can that muscle be strengthened with exercise?

This is a good question. Ptosis usually results from the levator muscle loosening or stretching.  Rarely it is from underlying weak muscle.  Neither case is going to improve with any muscle exercising.  The only real treatment is tightening the muscle to make it "stronger".  See an oculoplastic surgeon.

In ptosis, the tendon which is attached to the muscle gets stretched and the lid will droop.  There are no exercises you can do to strenghen the muslce.  The only option is surgical repair.   

Exercising the upper eyelid will not cure or correct your ptosis.  Acquired upper eyelid ptosis is often the cause of an issue with the tendon and not the muscle itself.  Opening and closing the eyelid is actually just as likely to further weaken the upper eyelid.  Surgery shortens the tendon to repair the ptosis.

Eyelid ptosis is usually a mechanical problem due to a weakness or detachment of the muscles that raise your eyelid. It can also be secondary to medical problems

There are a variety of causes of an eyelid droop. However, the only durable repair option is surgical. There are drops that can be used to raise the lid but these are used only in cases of a temporary eyelid droop (Botox related) or for in office assessment prior to surgery. I hope this information is helpful for you.Stephen Weber MD, FACSDenver Facial Plastic Surgeon

There are multiple causes of ptosis: congenital; acquired; neurogenic (nerve related)' aponeurotic (tendon related); myogenic (muscle weakness); and traumatic.  There are no known exercises to strengthen the muscle.  When it becomes visually bothersome or aesthetically displeasing then it is time for a surgical evaluation.  I hope this helps.