Tentative house rule for D&D:
Initiative. First round, roll Intelligence (alertness) Second, Dexterity (speed) All other rounds, Constitution (endurance)
Pros? Spreads the stats, allowing for a bit of tactics, and for not every combat to be about Rogues getting alpha strikes. Also, makes Con an active stat in the game!
Cons? What round are we on again?
Treehouse: 
Hmm, would like to see this tried out, but its certainly seems an interesting way to spread the load as you say.
That’s a good idea.
We haven’t rolled initiative every round for years but that may be because it wasn’t interesting enough.
What about randomly determining which to use?
Or letting the environment determine which to use?
Random might be a bit weird. Definitely feel the scene should direct the roll though. I can think of many occasions where Str might be best. Wis already sees use in the surprise round, and Cha would get a look in during social conflicts if you’re of a mind to use them.
Some kind of Round tracking dice, let’s call it the ‘Intensification Dice’?
Yes! To be honest, I’m blending bits of Next and 13A together in my min all the time. Sometimes by design, often by forgetfulness.
I really like the idea of doing this. It seems like it would give combat a certain dynamic flow to it. The opening rounds would be all about getting your bearings, then everyone rushes to try and deal a decisive blow before the opponent can. Then it’s all slogging through til the end from there.
I’ll try this out at a home game and see if it works! Next combat is swift enough that the extra bit of dice rolling every round won’t be too bad. We use notecards to keep track of initiative anyway, so I don’t suppose it would be that unwieldy.