By the book, the sickle isn't a finesse weapon. Would game balance suffer were I to house rule the sickle into a finesse weapon?
No, it does not.
The damage dice for the sickle according to the PHB (149) is 1d4 slashing damage and only property is being light. To put things into perspective is best to use the closest simple weapon that fulfill the finesse property, that is the dagger. The dagger do 1d4 piercing damage and has the finesse, light, and throwing properties. Given that both weapons have identical damage, it is safe to assume that the consequences of giving the finesse property would make no difference. But, to try to expand the answer.
Bypassing damage resistance
As far as balance is concerned this is one of the biggest points to address. Given that creatures can have resistance to one type of damage, the question would be, does it make any impact? Well, let see, first we have to analyse which classes would benefit for this. First, we take from the equation any class with the martial weapon proficiency, since there are better weapons for those classes, the monk class because can turn any monk weapon into a DEX based weapon, and any caster class that won't be hitting things on melee. That leave us with the Rogue, Druid, Bard, and Cleric.
Lets reduce a bit more, the druid can use Scimitars, a 1d6 slashing martial weapon with the light and finesse properties. A better alternative than the sickle. So, this leave us with the rogue, bard and cleric. The highest damaging weapon that the rogue and bard can use is the rapier, a 1d8 piercing weapon with the finesse property. And this is where we are going to take into consideration the damage resistance.
Let assume that we have an enemy with piercing resistance (that is, it reduces by half any piercing damage it takes).
Rapier: 1d8 + 5. The average damage of 1d8 is 4.5 so 4.5 + 5 = 9.5 points of damage. Applying the resistance would yield 4.75 points of damage on average.
Sickle: 1d4 + 5. The average damage of a 1d4 is 2.5 +5 = 7.5.
The difference is 2.75 points of damage per hit favoring the sickle. Nothing to write home about, though. That is, taking in consideration just the weapon attack. The real difference comes when you consider the rogue's Sneak Attack, where the damage reduction really hurt the overall damage outcome. The other classes do not benefit from this as much, since they or have some martial archetypes or their main form of attack is not melee.
Therefore, the only class that really benefits from this change is the rogue, but it has alternatives to solve this problems: Multi-classing and feats. Funny enough, the rogue is the class that has more ASI/feats than the norm, second to the fighter, and that really benefits from a bit of multi-classing. So, there is not really a big advantage to the rogue, specially if the player wants to maximize the damage it can do.
Retracting a decision
As the DM you have all the rights to retract a decision on your games. If a change suddenly seems more powerful than expected and its ruining everyone game, you can always take it back.