How do I add names to objects in JSON using PowerShell?

One way of tackling this: (You can copy and paste the full contents of the code box to try it out. I've commented what various steps are doing)

$ExistingJson = @"
{
    "Category": [{
            "TechnicalName": "N988AB1",
            "Name": "House"
        },
        {
            "TechnicalName": "H181AG3",
            "Name": "Apartment"
        },
        {
            "TechnicalName": "X123XY5",
            "Name": "Villa"
        }
    ]
}
"@

# Convert the json string to an object we can manipulate.
$ExistingObj = $ExistingJson | ConvertFrom-Json

# Create an empty hashtable for the new items we are creating.
$HT = @{}
foreach ($Category in $ExistingObj.Category) {
    # Match any digits, grouped as "name", after any character [A-Za-z] which are followed by [A-Za-z] - this is pretty basic and may need to be changed if your TechnicalName string changes, or you want different information from it.
    [System.Void]($Category.TechnicalName -match "(?<=[A-Za-z])(?<name>\d+)(?=[A-Za-z])")
    $NewItem = [PSCustomObject]@{
        TechnicalName = $Category.TechnicalName
        Name          = $Category.Name
    }

    # Add a new entry to the hashtable with the discovered digits by it's group name "name" and the object created above with the existing information as it's value.
    $HT.Add($Matches["name"], $NewItem)
}

# Create a new object with a Data property with the value as the hashtable.
$NewObj = [PSCustomObject]@{
    Data = $HT
}

# Convert it back to Json
$NewObj | ConvertTo-Json

Results in:

{
  "Data": {
    "988": {
      "TechnicalName": "N988AB1",
      "Name": "House"
    },
    "181": {
      "TechnicalName": "H181AG3",
      "Name": "Apartment"
    },
    "123": {
      "TechnicalName": "X123XY5",
      "Name": "Villa"
    }
  }
}