I'm in my mid-50s, and one lesson I've learned is that the lack of money is the source of a LOT of stress in life. I HATE it when I don't have enough money. I've lived too many years where blowing the motor in a vehicle or another unexpected event could have put me on the streets.
Waking up in the middle of the night stressed out about how you're going to pay bills is HORRIBLE.
So yes, money can be a huge factor in a person's happiness. Not having to worry about paying bills, not having to drive a junk vehicle that is a constant source of frustration, not having to worry about heating your home because fuel oil is outrageously expensive, being able to do what you want when you want to, being able to help your kids or grandkids get a good education - those things are priceless.
At the same time, a LOT of money can be a huge factor in a person's unhappiness.
As jgnat wrote, there is a point of diminishing returns. There is a great book, Your Money or Your Life, that discusses this in detail. They have a graph that shows a real increase in happiness as income expands and debt diminishes. BUT, at a certain point, once needs and legitimate and important wants are met, there is a rapid leveling off, and after that happiness actually DECREASES, even though wealth might increase.
I've told all my kids that, if there was ONE THING that they could do that would make their lives much, much better and happier, it was to learn how to handle money and stay out of debt.
I'm not sure they listened all that well!