Huh, I was thinking of axing the movement, but it does add something a bit more interesting. I should probably specify that it won't take you out of range of whatever triggered the reaction (e.g. if it was a reaction to getting hit, you still get hit even if you move out of range of the attack). Maybe I should remove the DEX bonus, turning it from a half feat to a full feat.

If I keep the bonus to DEX saves, I might only have it persist until you use a reaction, so it's either on or off, rather than trying to track how many reactions you've used. Or just make it a flat +1 or +2 that's always on.

Here's a revamped version:

Lightning Reflexes
  • You gain a +1 bonus to Dexterity saving throws.
  • You may take one additional reaction each round.
  • Immediately before or after using a reaction, you may move up to 5 feet. This movement doesn’t provoke opportunity attacks.

My other thought was to allow you to use a reaction to boost your DEX saves (instead of having an always-on DEX save bonus), like the War wizard's Arcane Deflection, but only for DEX saves. Here's what that might look like:

Lightning Reflexes
  • You may take one additional reaction each round.
  • Immediately before or after using a reaction, you may move up to 5 feet. This movement doesn’t provoke opportunity attacks.
  • When you fail a Dexterity saving throw, you may use your reaction to gain a +4 bonus to that saving throw.

(+4 is what Arcane Deflection gives, and it also only applies to just that one save, not to all saves. Ergo, it has to be strong enough to make it worth using.)