[RPG] By RAW, do monks include their ability modifier in the damage roll of their unarmed strikes

Yes: the logic works if you go from General to Specific.

But first, a word from the lead rules guru, Jeremy Crawford, from the 2019 Sage Advice Compendium:

Can a monk use Stunning Strike with an unarmed strike, even though unarmed strikes aren’t weapons?
Yes. Stunning Strike works with melee weapon attacks, and an unarmed strike is a special type of melee weapon attack. {italics/bold mine}

The general rule for "Melee Attacks" applies to all characters:

Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or similar forceful blow (none of which count as weapons). On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.(PHB, Ch 9; Basic Rules p. 76)

Check out the bolded part. Before any modification for the specific Monk case, we:

1. add the Strength modifier and proficiency bonus to the attack roll, because we are proficient with unarmed strikes, and then

2. add the Strength modifier to the damage (1) of an unarmed strike.

   The normal damage of an unarmed strike is 1 + Str Mod. The normal damage for any other melee weapon attack is {weapon damage} + modifier. The game then modifies this general rule such that ranged weapons use Dexterity instead of Strength, and attacks with finesse weapons can choose to use either Dexterity or Strength.

Look at the pattern

Go back to how weapon attacks work. Unarmed strike uses the same logic¹ - the unarmed strike is treated as a melee weapon attack even though it is also called "not a weapon" in the SAC cited in the beginning. In the original weapons table, Unarmed Strike was a weapon that did 1 bludgeoning damage¹. The Sage Advice Compendium table (page 10, Sage Advice 2019) clarifies the adding of Str modifier to the damage of melee weapons attacks.

The rules on damage rolls state:

When attacking with a weapon, you add your ability modifier—the same modifier used for the attack roll—to the damage. (Basic Rules, p. 77)

Melee attack damage is {base} plus {mod}:

Unarmed strike damage is {X} + {mod}
A melee weapon's damage {X} + {Mod}
A Dagger's damage is.... 1d4 + {mod}
A Halberd's damage is... 1d10 + {mod}
A Mace's damage is....... 1d6 + {mod}

That makes internal sense; unarmed strike is a melee weapon attack. Damage from a melee weapon attack takes the form of "weapon damage + Str Mod".

If I have a Strength score of 10, my unarmed strike is 1: 1 {+0}
If I have a Strength score of 12, my unarmed strike damage is 2: 1 +1.
If I have a Strength score of 8, my unarmed strike damage is 0 (1 - 1)
If I have a Strength score of 8, my damage with a Mace is (0-5): d6 - 1 ...

The normal damage of the unarmed strike is 1, plus the character's Strength mod (which can be a negative value). _{I have a vague memory of a youtube or early discussion about unarmed strikes pointing to a variety of unique issues - like you can't use them for most Paladin smites - but my current files don't have them}

Monk-Specific

From PHB Chapter 3, the Monk-specific modification of the above is from the Martial Arts feature, which all Monks get. Since the question isn't asking about Monk weapons, we will address the "you are unarmed " case.

You gain the following benefits while you are unarmed or wielding only monk weapons and you aren’t wearing armor or wielding a shield:

• You can use Dexterity instead of Strength for the attack and damage rolls of your unarmed strikes and monk weapons.

The normal damage of 1 + Strength mod becomes 1d4 + Strength mod (until level 5, when it becomes 1d6 plus Strength mod):

• You can roll a d4 in place of the normal damage of your unarmed strike or monk weapon. This die changes as you gain monk levels, as shown in the Martial Arts column of the Monk table.

QED. We went from general (use Strength) to Monk-specific (can use Dexterity instead), and the Monk subs in a d4/d6/d8/d10 (+ Str mod) for the normal '1' (+ Str mod) damage inflicted by an unarmed strike.

On a related point: the feat Tavern Brawler upgrades the damage that an unarmed strike does in bullet 3:

3. Your unarmed strike uses a d4 for damage (PHB, p. 170)

What is that an upgrade from? A d4 is the lowest damage die. The upgrade is from 1 damage to 1d4 damage, which is consistent with all of the above.

Bottom Line

When you are a Monk, you can add your Dexterity modifier for attack and damage rolls when making unarmed strikes.

• Note: Your Monk can, if they wish, use their Strength modifier instead! Yes, a Monk can do that! If Dexterity is 16, and the Monk finds and attunes a pair of Gauntlets of Ogre Power at level 3 (setting their Strength score to 19 and making their Strength modifier equal +4), the Monk can apply the benefit of +4 to hit and +4 to damage, since it is greater than the Dexterity modifier. The Martial Arts class feature says "you can" rather than 'You must!"

¹ In the original printing of the basic rules and the PHB, unarmed strikes were listed as melee weapons and included on the table of weapons (right below Spear):

In the first PHB errata in 2015, the designers removed "unarmed strike" from the weapon table and added the current text on unarmed strikes on page 195 quoted above - that text (redundantly) bakes the general rule for adding modifiers to weapon attacks into the rule for unarmed strikes. (Some attacks require weapons, - paladins' Divine Smites - and the devs apparently used this errata to prevent hand inflicted Divine Smites, among other things. Thus the SA comment about "special kind of weapon attack")

For What It's Worth: on D&D Beyond, the damage calculator adds in the ability modifier to the Monk's unarmed strike (Thank you @Medix2), and the pregenerated characters at WoTC's site also do that (thank you @Sdjz).