The 5th edition Monster Manual gives Bugbear stats for morning star and javelin attacks, but not for unarmed attacks.
I improvised 1d4 +2, but how much damage can an unarmed bugbear actually deal?
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On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
Now we need to know the Bugbear's proficiency bonus and we can look to page 8 of the Monster Manual which provides a lot of math behind the Monster therein. A chart shows that a CR of 1 provides a proficiency bonus of +2 which is consistent with all of its attacks and skill proficiencies (though note it does have expertise in stealth), and so we can conclude the following:
The Bugbear has a +4 to hit with an unarmed strike (+2 Strength modifier, +2 proficiency bonus) and it will deal 1 + 2 bludgeoning damage (the basic formula for an unarmed strike with a Strength modifier of +2).
Of note the Bugbear's Brute ability will not work with unarmed strikes as it states:
A melee weapon deals one extra die of its damage when the bugbear hits with it (included in the attack).
Unarmed strikes are not considered weapons and so this feature will not benefit them however the Surprise Attack feature will still work as normal:
If the bugbear surprises a creature and hits it with an attack during the first round of combat, the target takes an extra 7 (2d6) damage from the attack.
No part of this requires a weapon, only an attack and an unarmed strike is an attack.