The almost pure invincibility of the ships hull made me wonder why it wasn't more commonplace.
Live Oak is what you are thinking of.
White Oak was relatively common, cheaper, but also not as strong or long lasting. Live oak, for being much denser though is also usually a squat tree and can cause even more difficulty in harvesting enough even for framing a vessel. White oak grows nice straight and tall.
But to answer your question more directly. Yes it was understood to be a very valuable building material for its strength and aging. Its why Joshua Humphreys called for Live Oak to be used in the appropriate portions of all 6 of the Original Frigates of the US Navy, of which Constitution was just 1. Which combined with 3 of the 6 being what we might call a 'Super Frigate' intentionally up gunned and sized by Humphreys to outfight peer ships helped set Constitution, President, and United States up for success. We dont talk about the poor old Chesapeake. Though even the slightly smaller Congress and Constellation were built intentionally denser and stronger in the framing than peer ships, at the cost of some speed, weight, and of course cost.
But that meant surveying 1000's of trees along the Southern Coast, and sending 100's of men over the course of years into the swamps and coastal forest land to harvest them, including a large % of whom were slaves. Many of whom would die of disease or injury in the process, and at great cost and slow going.
But Live Oak was such a critical resource the US government even treated like a national security issue. Inspection surveys were done to assess the size of tree stands remaining, patrol boats funded to prevent illegal poaching, and even today the US Navy owns some stands outside Pensacola, and has other logs stored literally in big ponds and most recently was used in her rebuild in the late 20's. Though timber stored on land and harvested in the 90's was used in 2016 to rebuild her bowsprit among other sections.
And finally we should note its not 1 wood fits all, different parts of the ship have different requirements, for strength, weight bearing, flexibility, wear and tear, etc. So Live Oak would do part of the job, along with Fir, Cedar, Pine, and other Oaks depending on what could be had.
You may be interested in this older response of mine that even quotes some old reports by the Navy to Congress about their concerns over supply of the wood by the 1830's.
In addition to the really nice answer /u/DBHT14 provided, I'd also add a couple more points for you, regarding the supposed "cannonballs bounced off its hull" myth that we have. Adapted from this thread:
Regarding the issue of an 18-lb cannonball bouncing off the sides of the ship, I am skeptical, for a couple of reasons:
Constitution suffered casualties during the fight with HMS Guerriere, when the cannonballs were supposedly bouncing off, and in later fights. Not all those were caused by balls entering through gun ports.
The citation for this legend often references the ship's "25-inch thick" oak hull. The problem with that is the ship didn't and doesn't have anything resembling 25 inches of oak on the hull; its knees, braces and structural material might be that thick, but the hull is not.
The citation you commonly see online traces itself back to Toll's "Six Frigates," which takes its citation from Tyrone G. Martin's "A Most Fortunate Ship: A Narrative History of Old Ironsides," which was written in 1980 by a former captain of the historic ship and is ... florid ... at times, and which does not footnote its sources (they are grouped alphabetically by chapter).
In what is a bit of a historical oddity, the US Navy still operates a wild oak preserve inside Gulf Islands National Seashore. The preserve is managed for maintenance of the Constitution, the oldest US warship still in commission.