No.
The key here is that the Monk weapon ability doesn't actually add the keyword "finesse" to the weapon, it just lets you use Dex for attack and damage.
Defensive Duelist specifies that a weapon have that keyword, and thus, no, it can't be used.
Adding more fuel to this fire is Jeremy Crawford talking about natural weapons and coming down on the side that just because you are attacking with Dex does not mean you are adding finesse:
No. A weapon has the finesse property only if its description says so, and using Dex. to hit doesn't equal finesse.
Would this be a game breaking houserule? No, that's quite unlikely. The Staff does d6 or d8 damage, and there is already a finesse weapon that does d8 (Rapier). The monk isn't proficient in that, so it's not available, but it's not out of bounds for them to be able to use it.
Note: Brian points out that two monk weapons are finesse weapons, and as such would be usable with this combination (and get the Monk's proper monk weapon die). These are the dagger and the short sword.
No, there is no good balance reason
The designers may have imagined there was; it’s conceivable they considered the option of Weapon Finesse as your 1st-level feat (or Weapon Focus, Exotic Weapon Proficiency, Quick Draw, and so on) to be important advantages of full-BAB classes. I tend to believe it was more of an “imagery” thing, though – full-BAB classes were the “weapon masters” and so weapon-related feats tended to get this requirement. Not because balance demanded this, but rather because their image of what each class was “supposed” to be like. But either option is pure speculation on my part, nothing more.
The theory:
Regardless of what they were or were not thinking when they wrote the feat, the BAB +1 requirement on Weapon Finesse is not only unnecessary (just as it is not for Weapon Focus, Exotic Weapon Proficiency, or Quick Draw), but is also exceptionally problematic. That’s because, unlike the other feats I mention (with the possible exception of Quick Draw), Weapon Finesse is a build-defining feat – someone with Weapon Finesse is going to have low Strength and high Dexterity, and is going to be interested in melee combat. Thus, they need Weapon Finesse to function correctly.
Worse, this description, of a high-Dexterity, low-Strength melee combatant, does not describe most full-BAB classes – you can make a Dex-based fighter, certainly, and even a Dex-based barbarian with the right supplemental material, but these are not the norms for those classes. Fighters get proficiency with heavy armor and shields as a major class feature; you cannot use those on a Dex-based build. Paladins are worse – same proficiencies, but far fewer feats. And rangers are worst of all – their ability to ignore the Dexterity requirements of their Combat Style feats is a major class feature, one of the only ones that is both relatively unique and relatively useful for that class.
And then you have medium-BAB classes that, unlike the full-BAB classes, are often perfectly described as high-Dex, low-Str melee combatants. Monks and rogues wear light armor if they wear armor at all, have ¾ BAB, and find themselves in melee. Rogues in particular see a lot of advantage from high Dexterity: requirements for Two-Weapon Fighting feats, associated score for critical skills. These are characters who want to build around Weapon Finesse.
Which sucks if you are level 1, and have BAB +0. You are forced to sit through two whole levels of hideously poor accuracy, just so you can make the math work by taking Weapon Finesse at level 3. This doesn’t add anything to the game, it just makes those classes suffer needlessly.
The practice:
The BAB +1 requirement on Weapon Finesse (and most other feats, honestly) is waived in the overwhelming majority of games I play in (the only games where it isn’t, that I can recall, are those in which no one wanted it and therefore didn’t bother asking the DM), and every single game I DM. In fact, most of the people I play with forget that it has that requirement. And our games are better for it. It allows us to play at low levels without being forced to suffer through unnecessary glitches like those caused by Weapon Finesse’s requirement.
I strongly recommend every DM to waive that requirement. I’d also recommend doing so on other, similar feats (certainly each of the ones I listed earlier). To not do so, I think, is a mistake. A mistake that the authors of Player’s Handbook made, true, so an understandable mistake, but a mistake nonetheless.
In fact, I would echo Thales Sarczuk’s suggestion, and strongly consider just making Weapon Finesse a free option.
Best Answer
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.