If you have a weapon or something similar in your hand after you've used your Action for whatever purpose, you can attack with that weapon once as a Bonus Action. My question is if that attack could be an Unarmed Strike, because I don't see any reason why using a dagger and using a fist would be different in that regard. 

Quote from Helllopwpep>>

If you have a weapon or something similar in your hand after you've used your Action for whatever purpose, you can attack with that weapon once as a Bonus Action.

What gives you that idea?

Here is the rule for two-weapon fighting: 

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative.

Unarmed strikes do not have the "light" property and therefore cannot be used as the bonus attack under the rules as written.

This might be something your DM would allow through homerules, but you would have to get their sign off and it would not be the official rules.

Rules As Written: This is a class ability of the Monk. Other classes must use two light one handed weapons. If they have the Dual Wielder Feat then the weapons don't need the light property. But a fist is not considered a weapon. 

But, as always, the final ruling is up to the DM of the table you're playing at. 

I believe excluding Unarmed from TWF was a deliberate design choice to make it a Monk specialization. Plus, unlike weapon attacks, Unarmed Attacks don't actually require a hand be dedicated to it, you can use any body part for the blow. So theoretically you could sword and board with a shortsword or scimitar and then make an unarmed attack while still holding the shield, which is obviously against the intent of TWF and would need a bunch of extra language for something that pretty much no class besides Monk (who don't have the prof and are penalized for using shields) would be inclined to do in the first place.

This is RAW, but, as a DM, I would personally always allow a player to be able to punch with their fist or kick an opponent as a bonus action, even if they were NOT using a light weapon.  Does it physically make sense?  That's the question.  Can I swing my heavy two-handed sword at an opponent and then try to kick them?  Sure!   Maybe the kick is with disadvantage because the heavy weapon throws you off balance.   Same if I were to fire a two-handed bow and arrow and then try to strike a nearby opponent with the end of my bow melee style ala Legolas.  Again, disadvantage on the melee strike, but still ...  Mind you, it's kind of a waste of a bonus action.  Unless your a monk, the damage you would do with a punch or kick is just 1+STR mod, so at the most 6.  But, if a player doesn't have anything else to spend their bonus action on, and they're in the middle of combat and want to add something ... why not?
Of course, I always tell players at the table .. if YOU can do it, the enemy can too.  That's the best way to balance these added combat rules on the fly.  

Also, LOL ... "Unarmed strikes do not have the "light" property" ... True, but if any weapon has the "light" property it would be my empty fists, right?  I mean, cmon.

You do you at your table, but games need to have rules and the monk needs something to make it attractive and unique.

Rathkryn: Dual Wielder eliminates the requirement the weapon be light, not the weapon be one-handed (or not two-handed)

Sure, but I don't think allowing players to make an unarmed attack as a bonus action is really taking away from the Monk.  Consider the Monk's benefits ... You can use Dex or Str for the attack.  Your damage is a D4 plus Dex or Str, so maximum of 9 damage (with higher damage at higher levels).  That's still a benefit.  And when you add on to that Ki-strikes and everything else ... Honestly, Monks are pretty awesome.    And really, if you make the non-monk player's attack an attack with disadvantage, because they don't know martial arts ... really, I think that pretty much balances the whole thing out.  

But, I'll be the first to admit, allowing something like unarmed bonus action does break the rules.   For the sake of story or a player feeling cool though, I'm always fine with a little rule breaking.  

Quote from Thatcher_Tydes >>

Sure, but I don't think allowing players to make an unarmed attack as a bonus action is really taking away from the Monk.  Consider the Monk's benefits ... You can use Dex or Str for the attack.  Your damage is a D4 plus Dex or Str, so maximum of 9 damage (with higher damage at higher levels).  That's still a benefit.  And when you add on to that Ki-strikes and everything else ... Honestly, Monks are pretty awesome.    And really, if you make the non-monk player's attack an attack with disadvantage, because they don't know martial arts ... really, I think that pretty much balances the whole thing out.  

But, I'll be the first to admit, allowing something like unarmed bonus action does break the rules.   For the sake of story or a player feeling cool though, I'm always fine with a little rule breaking.  

Maybe say they can unarmed strike but their proficiency bonus wouldn't apply to the attack since they're not monks that have BA unarmed strike by default. It's your game, I'm just brainstorming.

Even with that, unarmed attacks' damage is just their STR mod +1 , so unless they have a 0 STR modifier or higher, their unarmed strike would just be spicy narrative flavor.

Even better!  Yeah, I like the idea of not applying prof bonus.  And yes, spicy narrative flavor for sure.  

Something like "You slap the taste out their mouth" or "You kick him in the shins, now he's even more mad at you"

Quote from Thatcher_Tydes >>

True, but if any weapon has the "light" property it would be my empty fists, right?  I mean, cmon.

Unless you're heavy-handed? 

I'll .. see myself out =)