RAW
No damage dice => No dice to double
Sucks, but them's the rules. So a critical hit with an unarmed strike does nothing more than 1 damage, excluding modifiers. (I'll be excluding those for the rest of the answer) The tables in the player's handbook doesn't list a "1d1", just 1 damage. "1" is not a dice roll, it's a number.
Of course, the designers do seem intent on making unarmed strikes do 1 damage. I doubt it's a typo, especially considering the basic rules have been updated (it's on v0.2). They even supplied a feat (tavern brawler in PHB) and a class (monk) which increases the unarmed strike damage from 1+mod to some actual dice (like a 1d4). If they never intended the damage of unarmed strikes to be so lame, why would they provide that feat and that class?
As a side note: a critical hit from any weapon causes two failed death saving throws... including unarmed strikes. This is possibly the only time they are on par with more traditional weapons.
Possible RAI and RAF; The Spirit of Critical Hits
Critical hits seem to be meant to do more damage than a normal hit. This is reflected in the doubling of an attacks damage dice. It makes you generally do more damage than normal, although there is a slim chance that you don't do as much on that strike. You can see the odds for some weapon dice values here.
If you're looking to extend this logic to critical hits, I suggest instantiating a house rule for upping it to a 1d4 or supply 2 damage on a critical hit. This could be a fun rule, and it may alleviate the disappointment your players feel while getting that critical hit while hitting something with their... fists? ("Unarmed Strike" is really vague!)
However, it does seem that the designers really did want unarmed strikes to not hit hard unless you've had some experience. Once again, this is because of the Tavern Brawler feat in PHB and the monk class. Is this because they want people to use weapons unless they're a monk or a tavern brawler? Were they afraid of things like the grapplemancer builds coming into 5e? Hard to say. You could ask wizards of the coast by emailing sageadvice@wizards.com.
A Note
5e greatly acknowledges the importance of house rules in D&D culture, so I would not stick with the RAW reading of this. In that article linked above, written by Jeremy Crawford, one of the two lead designers for 5e, says:
We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players.
The RAW just is not fun and takes the joy out of a critical hit. Come up with a rule that seems right (doubling to 2 damage, or going to 1d4 damage, or something else) for your group. In the words of Jeremy Crawford:
Fifth edition now belongs to the thousands of groups playing it.
Modify 5e to have fun in your group!
Only the part that is contingent on the attack roll itself is doubled
There seems to be intent that it is not doubled, as evidenced by the Green Flame Blade question below:
In a tweet Crawford specifically answers a question about the subsequent damage on Green Flame Blade.
Question:
If you critically hit with the attack part of Green Flame Blade, and
are greater than level 5, do you roll extra dmg against the 2nd
target?
Answer from Crawford:
The splash damage of green-flame blade isn't affected by the attack
critting. Think of the attack as process X & the splash damage as Y
Coupled with Sage Advice 2020 p13:
Can spell attacks score critical hits?
A spell attack can definitely score a critical hit. The rule on
critical hits applies to attack rolls of any sort.
I do realize that JC's Tweets are no longer dogma but until definitive clarifications come from Sage Advice they will have to suffice. That said JC's Tweet on Witchbolt criticals dovetail nicely with Melf's Acid Arrow in so far as the adjudication of damage on subsequent turns.
Question:
What if the Witch bolt crited? Since there's no attack roll for the
subsequent damage, would it roll only the first as a crit or all of
them as crits? (I assume only the first one, but eh)
Answer:
Witch bolt's initial attack can, indeed, crit, but the damage
delivered via actions on subsequent turns is unaffected by the crit. A
crit isn't intended to affect the damage of other
actions/reactions/attacks.
These lead me to believe that the only part of a spell attack that is doubled is the part that is dealt as part of the actual attack roll. As the subsequent round's damage would be the 'Y' or even 'Z' in this instance.
I went through the podcasts and couldn't find anything else, mostly the spells section dealt with targeting in the one from Jan 17, 2017.
Best Answer
1 and 3
PHB, pg. 196
Crits only double the dice rolled, not the modifiers. It also doubles any and all dice applied to the damage. If a spell or ability added any dice to the attack damage, you roll those twice as well. So, a +1 dagger does 2d4 + 1 damage on crit. Another example, Eldritch Blast hitting a target affected by Hex does 1d10 + 1d6 normally and 2d10 + 2d6 on crit, plus any relevant modifiers such as Agonizing Blast.
2
When damage is listed for any weapon, attack, spell, etc it applies to all of the dice and modifiers that come before the damage type. Thus, a weapon that does 1d6+2 slashing does all slashing. A weapon that does 1d6 slashing +2 necrotic will do 1d6 slashing damage and 2 necrotic. If it were listed as 1d6 +2 necrotic, it will do all necrotic damage. Damage types are never assumed, they are explicit.