Soft 404
Nice. I like the lamp shade, nice touch. Just needs the pull cord switch 
It was suggested here that you may be able to get down to 200mV:
May be able to fit that toroidal core in the base ?
Excellent reference! Karen did a wonderful job on her joule-thief circuit. I thought about mentioning it, takes for mentioning Karen's project!
These modules are beginner friendly and provide a regulated stable DC output. So you could also run a neopixel and small mcu off a single alkaline cell.
Yes, past 600 mV there isn't much left... Of course you can use a fresh cell too
Some of the low cost dollar store solar lamps use tabbed ni-mh cells, I'm sure if I were to shop around I could find them for pennies...
Every mW counts! Dave Jones claims you can fly to the moon on such power budgets
Time to stock up on 5V solar panels just in case: Checkout this test rig