Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from Houston to Patagonia as you normally would.
Yes, there are multiple flights from Houston to Patagonia for under $1,700. The cheapest flight booked recently is on Multiple Airlines for just $1,689, but on average you can expect to pay $5,155.
There are currently 5+ open flights from Houston to Patagonia within the next 7 days for less than $2,700.