Let's assume a sidekick with the prodigy role, CHA 20 and the potent cantrips-feature :
"The sidekick can add its spellcasting ability modifier to the damage it deals with any cantrip.")
Could this Sidekick add the spellcasting ability modifier of +5 to each beam of an Eldritch Blast or does the sidekick have to choose a single of the beams that gets the additional damage?
Each successful hit with eldritch blast deals 1d10 force damage. The Potent Cantrips feature allows the Sidekick to add its spellcasting ability modifier (in this case CHA) to the damage dealt by a cantrip. This means that instead, each hit will deal 1d10 + CHA force damage.
Each beam fired by the spell deals this damage, and so each beam will benefit from the bonus to that damage. This is somewhat similar to the way that magic missile works (except that in our case, we do roll the damage separately for each beam), which is why it also allows for multiplicative scaling of damage bonuses that aren't more restrictively worded.
Potent Cantrips says:
The sidekick can add its spellcasting ability modifier to the damage it deals with any cantrip.
The trouble here is the phrase "the damage it deals". It is unclear what is meant by this phrase, as it can reasonably mean one of the following things:
Eldritch Blast unambiguously deals damage once for each beam thrown, so it is reasonable to call each instance of a beam dealing damage as "the damage it deals", but at the same time, it makes sense to say that the sum total of the damage dealt by all the beams is "the damage it deals".
But this confusion is obviously why you're here. We cannot nail down a ruling here as the rules are just ambiguous. So the next best thing I can give you how to handle making a ruling on your own.
The most important thing is to just be consistent. Make your ruling and stick to it. It can get frustrating when a DM makes contradictory rulings about the same feature, so we just need to be consistent in how we rule this one at the table. Now, being consistent does not mean the ruling should never be changed. Suppose you rule that you add the modifier to every beam, and it turns out that the ability seems more powerful than it should be, consider changing the ruling (and involve the player in the discussion). Similarly, if only adding the modifier once is too underwhelming, consider changing the ruling. Either way, just be consistent once you have decided what the ruling will be.
If you care, Jeremy Crawford did offer his unofficial ruling on a feature with the same wording, the Arcana Cleric's Potent Spellcasting feature:
Starting at 8th level, you add your Wisdom modifier to the damage you deal with any cleric cantrip.
When asked how this interacts with green-flame blade, JC responded:
Potent Spellcasting adds to dmg. dealt by a cantrip. GFB: add to 2nd target, then to both targets at 5th lvl.
So JC says you add the modifier to each instance of damage dealt by the cantrip. Again, Jeremy Crawford's twitter rulings are not official rulings.
The ability you are citing indicates it adds the spellcasters ability modifier to the damage they deal with any cantrip. Compare that to the agonizing blast invocation a warlock can take:
When you cast eldritch blast, add your Charisma modifier to the damage it deals on a hit.
That ability adds damage to each beam that hits. But the potent cantrips feature only says that adds to the damage it deals from a cantrip. Nothing in the text indicates it can add that damage multiple times.
On the flip side though, it doesn't say you can only add bonus damage to a single beam. So in theory, if you have a +5 stat modifier and two or more of the eldritch blast beams hit, you could divide that extra 5 points of damage between each hit however you wish.
