A bard casts Dissonant Whispers (PHB, p. 234) on a monster (diagram below). The monster has to choose between 2 paths. The left path is "safer" than the right path since his allies are that way but the right path led him a little farther from the bard.

Can the monster (A drow in armor and mid melee with at least 16 AC) choose the safer path or does he have to take the right one?

It's up to the DM...

Dissonant Whispers only requires that the creature:

must immediately use its reaction, if available, to move as far as its speed allows away from you. The creature doesn’t move into obviously dangerous ground, such as a fire or a pit.

From a pure RAW perspective, as long as it moves away from you and not into obviously dangerous ground, then it's following the spell requirements.

Two Paths

The spell itself is causing the creature terrible pain that is trying to escape from:

You whisper a discordant melody that only one creature of your choice within range can hear, wracking it with terrible pain.

It's trying to get away, but it's not 'frightened' of you or have any other effect. If the creature has any semblance of intelligence, it's going to run away from you towards its allies.

If it has no intelligence or knowledge/wherewithal to go towards friends, then it may be a toss-up.

But both options are firmly under the control of the DM as the spell itself doesn't give any direction here other than to move away from you.

Leave it to the DM to decide

Dissonant Whispers reads (in part):

On a failed save, [the target] must immediately use its reaction , if available, to move as far as its speed allows away from you. [...] The creature doesn’t move into obviously dangerous ground, such as a fire or a pit.

The location of the allies doesn't matter

The presence of allies down one path has no bearing on which way the target runs, according to the rules of the spell. The creature is said to ignore environmental hazards, but it is not said to consider the location of nearby allies.

How to parse "as far as its speed allows away"

The rules say "to move as far as its speed allows away from you." The wording "as far [...] away" could be measured as the crow flies, which would mean the right path in your scenario because the creature's speed allows it to move slightly farther away measured thus.

But it could also be interpreted to mean "the target uses ALL of its speed to move in a direction AWAY from you." Given that the target has a lot of allies on one side, it would logically choose to run down the path that puts it nearer to them (all other options being roughly essentially equal, distance wise).

Linear distance versus range distance

The above assumes that the paths prevent line of sight by the caster, which would require the caster to pursue to reacquire line of sight. Essentially, the linear distance travelled is equal either way, though one may be closer "as the crow flies."

If the linear distance traveled via the left path is the same as that of the right path but does not break line of sight, then the right path would probably parse as the only viable option.

It's the DM's call

Nothing in the rules dictates the internal logic of the path of the creature beyond moving "away from the caster." So, ultimately, it comes down to how the DM wants to interpret the rules. Either result is fine given the directive of "Rule Zero:"

• The DM narrates the results of the adventurers' actions.

The monster is not required to take a certain path

As per the spell:

On a failed save, it takes 3d6 psychic damage and must immediately use its reaction, if available, to move as far as its speed allows away from you. The creature doesn't move into obviously dangerous ground, such as a fire or a pit.

The monster only needs to use all of its speed, and move away from you. This even means that by a strict reading of the rules, a monster could "waste" speed by moving through difficult terrain or climbing; albeit, certain players/DMs might call "shenanigans" on such an interpretation.