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Say my players blinded a goblin, through either magical or non magical means, and the goblin knew that there was someone within 5 feet of it from before it got blinded. That goblin decides to flail around wildly in the hopes of hitting something.
How should I handle this goblin's attempt to attack? How should I roll to see if any players within 5 feet get hit?
A blinded creature must still state where (i.e. which direction/square) they are aiming their attack towards.
Typically, the goblin would know where the players are unless the players have taken an action to Hide. In this case the goblin could still make a Wisdom(Perception) check to determine the players' location.
If instead, for Rule of Fun, you want your blinded goblin to panic and, as you say, flail around wildly, then you could simply roll a d8 to randomly determine the square he attacks!
If the square the goblin attacks is empty, then the attack obviously just misses. Otherwise, as per the Blinded condition, the blind creature makes the attack at disadvantage.
Unless the target is hiding from the Goblin, the Goblin will still be aware of which square the target is in, which means that they can still target that person. Because of the Blinded condition, the Goblin will be forced to attack with Disadvantage.
If the target is completely hidden (by making a Stealth check and beating the Goblin), then it will be entirely unaware of the target's location. It might decide to attack the last position the target was in (which will be an auto-miss, if the target left that square, or an attack with Disadvantage if it did not) or you might decide it will do something else.
Realistically, the odds of a monster hitting someone if it doesn't know which square to target are effectively nil, so the quickest way would be just saying the monster flails around without hitting anything, but you could roll a die to decide which square it targets if you feel that would add something to the encounter.
Attacks need a target. The goblin needs to specify the target. Being blind does not render the goblin incapable of knowing there is an enemy adjacent to them. The goblin attacks its target at disadvantage.
In order for the character to be undetected by a blind goblin, the character would have to be hiding. Actively trying to conceal their presence by sound, motion, etc.
The rules clearly cover what being Blinded does in mechanical terms:
Blinded
A blinded creature can't see and automatically fails any ability check that requires sight.
Attack rolls against the creature have advantage, and the creature's attack rolls have disadvantage.
A character is still aware of the surroundings, even if it can't see them. This is effectively a codification of Unseen Attackers and Targets:
This is true whether you're guessing the target's location or you're targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target's location correctly.
The DM will need to determine if the Blinded character is aware of the location of an enemy. It's far simpler to allow characters to be aware of any adjacent character who isn't actively trying to hide. If the blinded character has to move to reach somebody, or is attempting to make a ranged attack, then it would be appropriate to have them guess a square.
In the case of NPCs, your bigger problem is that the DM has perfect knowledge - the DM knows where all the PCs and NPCs are and can't really "guess" without the influence of that metaknowledge. In that case, it would be appropriate to make some sort of random roll. On a grid, a d8 works well for picking a general direction (assign "1" to a particular square, usually "forward", and count around). For targets further away, unless they're doing something to be really loud, it's usually just faster to say the Blinded person just can't tell.
If you can find a story reason to swing things one way or the other, go for it, but do try to remember that D&D is not a reality simulator. You're telling a story about heroic (or maybe not-so-heroic adventurers). Delving into the minutiae of one goblin's blind swing is not likely to be worth the time - make the attack with disadvantage and move on to the next creature's turn.
D&D3.x had the concept of "Miss Chance". The exact roll could vary slightly based on the effect causing it, but it was a percentage of hit-or-miss that was pretty much unaffected by anything else - character skill was not a factor.
D&D5E does not have a similar rule.
As already pointed out in the other answers, a blind creature can still attack, only it will be at disadvantage. That much is set in stone. But I don't think your main concern is on the mechanics of blind-fighting, so I'll add to "how to decide where to attack".
In the case you presented, you want to know how the goblin would decide which square to attack, since he knew a foe was within 5ft. But first you should clarify if, before being blinded, (1) the goblin knew precisely where the foe was or (2) it had a general idea that the foe was somewhere within 5ft. From your question, that remains unclear.
If (1), then, in its next round, the goblin would attack the square where the foe was last seen. Blind-fighting rules applied, if the goblin still managed to hit something it considered a foe, then it would keep attacking that square until it misses.
In case (2) or after missing in case (1), the goblin would blindly pick a square within 5ft to attack. If it ever hits a foe, it would keep attacking that square until it misses, otherwise it would blindly pick another square and so on.
To simulate the blind pick, you just assign numbers from 1-8 to a 3x3 grid, centered on the goblin (see below), then roll a d8 and attack the corresponding spot. However, if the goblin has just attacked a spot and missed, it would try its luck on the next attack on a different spot, so you eliminate the previous spot by rerolling the d8 if you get the last number again.
Grid:
1 2 3
8 g 4 --> g = goblin
7 6 5
Example:
Roll a d8. Say you got 5, so you attack that square. If it's a miss, roll a d8 for the next attack. If you get another 5, which is the spot the goblin just missed and assumed nothing was there, reroll the d8 until you get another number. Say you got 2 and managed to score a hit. Next attack you won't roll a d8, the goblin would simply attack 2 again, until it misses. Repeat while the blindness lasts.
Note: both attacking and defending creatures' size and the attacker's reach impact the grid. Make sure you account for that in different scenarios.
Of course, you can adjust that to your needs. For instance, if the goblin knew a foe was somewhere 5ft behind it, you can determine the front and back of the goblin and then only assign numbers to the 3 squares behind it and roll a d3.
