Disagree with the people who say it's like a spell. Sure, barbarian HP "could" be a spell, but this isn't like any spell. You can use it any number of times during a round up to your attacks that hit and also cast it on other people's turn when you hit with an Attack of Opportunity. It's only used after you hit, and at that point you determine how much you want to add, if any. It stacks with all those nice smite spells. It does better damage than any other low level slot (2d8 to 3d8 without to hit roll or save) but doesn't do it's scaling as much with high level slots, but rather with more attacks. Sorry, but any of these don't fit 5e spells, all together just say "this is not a spell" in big blinking pink ASCII letters five feet tall.
Anyway, to break it out, between this, extra attack, and the extra damage to every attack gained later are what give the paladin his striker capability, with divine smite being a large portion (and possibly the largest portion if you have bonus attacks from another source like a feat or someone else hasting you).
If you take this away, you'll be taking a good chunk of their offense. If you make the smite spells free actions to cast instead of bonus actions you can make up for it to a lesser degree, but that's still once a round and not on AoO (or not during the round and hope for an AoO).
If you want the same balance, you need a daily-limited boost the the paladin damage output. And if you're doing that, it makes me wonder why you took it out to just put it back in.