I've considered something like this myself in the past, but abandoned the idea because it adds an unnecessary level of complexity and, more importantly, can produce more nonsensical results than the system in place.
For example, let's say that a character with an excellent Dex save takes the Dash action and moves their full movement (they cannot move another foot under their own power). Then a fireball goes off so he rolls a Dex save of 20+ and moves another 20 feet (escaping the fireball). Now another fireball goes off and he rolls another 20+ Dex save and moves another 20 feet. He's now moved considerably more than he ought to have been able to. Fireball becomes a poor man's mass Expeditious Retreat. As long as they have hp and you have Dex save spells, you can make your targets move.
Now sure, you could make that Dexterity save a reaction, limiting it to once per turn, but now you've changed the dynamic completely. Already taken an opportunity attack this round? Sorry, you auto-fail that Dex save. Already tried to save against one fireball this round? Sorry, that second fireball is going to deep fry you. A group of Flame Skulls becomes even deadlier than they currently are. Meteor Swarm becomes basically an auto-fail (four 40' radius spheres) due to its size. This means that Dex-based AoEs receive a significant boost, which ought to be factored in by reducing the damage of all Dex-based AoEs.
It also begs the question of why a character can move when targeted by Lightning Bolt or Fireball, but not Shatter or Cone of Cold (because the latter two are Con saves). So you'll probably want to change those too, as well as a lot of other spells and abilities that fall into the same category.
I devised a simpler method that I prefer to the aforementioned one, although I've never tried it because I still feel the added complexity might outweigh the benefits. If you are adjacent to the center/origin-point of the AoE, you have disadvantage on your save. If you are adjacent to the end/edge of the AoE, you have advantage on the save. Relatively simple, but given the amount of time it typically takes at my table to adjudicate AoEs (rolling multiple saves, tracking who made it and who didn't, applying full and half damage to the relevant parties) any added complexity is a hard sell for me.