My gut reaction to this was that the discarding of lower-than-previous-level totals would give the dice a substantial bias to the upside and that by the time players got to high levels, they’d have a lot more hitpoints than if they’d used the rules as written.
So I did some analysis. I rolled hitpoints for 600,000 9th level fighters. (Boy, is my dice throwing hand tired. And my d8 is worn down to a nub!)
I limited it to 9th level, since after that the number of hitpoints added is a constant. I limited it to fighters because I didn’t want to complicate it with multiple dice types.
200,000 characters used option 1: The rules as written: d8 every level, cumulative, for 9 levels.
200,000 characters used option 2: The rules as written, except max hp at first level.
Finally, 200,000 used Option 3: The house rule described by OP in this thread.
Average of Option 1 over 200,000 characters: 40.5
Average of Option 2 over 200,000 characters: 44.0 (No surprise there.)
Average of Option 3 over 200,000 characters: 42.4 (Seriously!?)
I thought it must be a fluke, so I rolled another 600,000 characters and then another 600,000 after that. (After which, I rolled another 600,000.)
That’s 2.4 million characters rolled, and the averages for groups of 200,000 per option always came out to within .1 of the results above.
Since the expected values turn out to be almost identical, and since this method prevents people getting screwed for long and also keeps them from getting too much long term advantage from lucky rolls, I think I’m adopting it. (I might also give them a 2hp bonus at level 8 in order to even it out perfectly with Option 2.)