I'm a rules guy. To me, taking 20d6 damage you could have easily avoided is punishment enough.
To me, the more serious underlying problem is that the player seems to want to play a fantasy supers game, and the DM wants to run one based on fantasy realism. IMO, this is the issue that needs to be resolved, not changing the rules on falling damage.
I thought one of the earlier suggestions was pretty good: ask the player to narrate how the character "only" took 20d6 damage when he reached the bottom, whether it is hitting the ground so hard at the point of impact that it slowed the character's fall, or clawing at the side of the cliff to slow his descent while laughing maniacally at the adrenaline rush.
I hear you. But let me make this more clear about the situation - The player does not want a supers game - he wanted to use his hit points as an elevator - a short cut. He offered no other justification and flat out said as much. I clearly explained the consequences that would be incurred and he chose not to do it.
What's interesting is that this player sometimes complains about how easy 5e is and how he wishes the game were grittier.
Now, if the player said his PC, "Looks over the side of the cliff. That's a long drop, but there are a few branches and roots and ledges. I think I see a way for me to get down. I'm going to take some heavy bruising and bashing, but I think I can make it!"
That would make a difference to me.
A lot of you disagree with that.
That's cool. As I said up-thread, after thinking about it, I'm good with this being a gray area of DM's call in my home campaign. This thread was helpful in reaching that conclusion.
BTW - I run very much RAW at AL and Cons.