What could cause a sudden spasm of my lower eyelid under 1 eye? It's not due to tiredness or stress as far as I know. Also have had daily headaches.

Myokymia. most common causes of lid fluttering are stress, fatigue, caffeine intake, rarely eye strain or hyperthyroidism. Usually the problem is self limited and resolves spontaneously. Occasionally it may be a precursor of benign blepharospasm or other disorders that respond to Botox therapy.
Visit MD. You should see your doctor. You may need to visit a neurologist for your head pain symptoms, if your physician cannot comfortably evaluate these symptoms. Do you have a history of migraines? Any early morning vomiting? Any dizziness? Keep log of headache symptoms and present to your physician. Feel better!