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One of my players in D&D 5e asked the Blacksmith if he could create some kind of small cushion to muffle the armor and not get disadvantage on stealth anymore. Is this possible? If so, how much does it cost?

There is such a thing, but it's not because of cushioning. It's called Mithral armor.

Mithral is a light, flexible metal. A mithral chain shirt or breastplate can be worn under normal clothes. If the armor normally imposes disadvantage on Dexterity (Stealth) checks or has a Strength requirement, the mithral version of the armor doesn't.

It can be found in the DMG under Magic Item Table B. As for the cost, that's entirely up to the DM but I would not make it cheap. Removing that disadvantage is a game changer for some.

For some examples on pricing:

• In the DMG, an Uncommon magic item runs between 101-500 gp.


• However in Xanathar's, under "Example Downtime Activities / Buying a Magic Item" it's a bit pricier:
  
  
  
  
  • Finding the item to purchase requires at least one workweek of effort and 100 gp in expenses as a start.
  
  
  • Then the actual item, for being Uncommon rarity is (1d6 × 100) gp, so up to an additional 600 gp.
  
  
  
  

A way to solve that problem without adjusting the armor

Take the Medium Armor Master feat

You have practiced moving in Medium Armor to gain the following
benefits:

Wearing Medium Armor doesn’t impose disadvantage on your Dexterity
(Stealth) checks.

When you wear Medium Armor, you can add 3, rather than 2, to your AC
if you have a Dexterity of 16 or higher. (PHB, p. 168).

It will also work with Half Plate armor when you can afford it, or when you acquire it.

Half Plate

AC: 15

Stealth: Disadvantage

Weight: 40 (PHB p. 145)

The idea behind stealth disadvantage is that armor is both noisy and REALLY FREAKING HEAVY. As someone who owns a scalemail shirt in real life, being light on your feet is much harder when you're wearing 45 pounds of steel scalemail. Adding more clothing really wouldn't help.

Mithral is a metal that is crazy light and hard as steel. It's also very rare and thus mithral armor is going to be expensive and hard to find.

Additionally, as a DM, you can make them go to someone who enchants items, and they can have it enchanted to not have stealth advantage, but 5e isn't very big into magical items, so it'd probably be a quest to find an enchanter, and it'd probably be just as expensive if not more expensive than buying mithral.

It is probably not practical to modify an existing suit of scale mail to avoid having disadvantage on stealth checks. However, a well stocked blacksmith probably can provide a mundane solution that has a similar effect, if your character has gold to spend and doesn't mind switching to a completely new set of armor.

A breastplate provides the same AC as scale mail, and does not impose disadvantage on Stealth checks since it is much lighter. It's quite a bit more expensive than steel scale mail (400 gp rather than 50 gp), but it is probably cheaper and easier to find than mithral scale mail or some other kind of magical armor.