Post-errata, page 195 of the PHB contains the text:
On a hit, an unarmed strike deals bludgeoning damage equal to 1 + your Strength modifier.
Moreover, page 196 says:
When attacking with a weapon, you add your ability modifier — the same modifier used for the attack roll — to the damage.
Presumably, an unarmed strike is a melee weapon attack, and thus by the rules on page 194 the ability modifier is the Strength modifier. Combining the two rules, this means that the final damage dealt by an unarmed strike is 1 + Str modifier + Str modifier, which seems a little... odd.
I can think of the following possibilities to explain what's going on here:
Adding the Strength modifier twice is the right thing to do.
Saying that the damage is 1 + the Strength modifier is just a reminder of the fact that you have to add the ability modifier, and thus you only add the modifier once when calculating damage. This is slightly supported by the fact that, pre-errata, unarmed strikes did just 1 damage. (If I'm interpreting it correctly, this also seems to be the conclusion reached in this question, though the reasoning there isn't exactly clear.)
An unarmed strike is neither a melee weapon attack nor a ranged weapon attack, and thus the ability modifier on the attack roll is zero, and thus the Strength modifier has to be explicitly included in the damage definition in order for stronger characters to deal more damage.
So, which is it?