jq: Can I use the name of an argument in the code itself?

If you remove the ($name): JQ will use the name as the key fallback:

$ jq -n --arg name value '{$name}'
{
  "name": "value"
}
$

From the jq Frequently Asked Questions:
^{Notable Differences between Versions}

: In which version was the abbreviation {$x} for {"x": $x} introduced?

Version 1.5

From the Object Construction ({}) documentation:

The value can be any expression (although you may need to wrap it in parentheses if, for example, it contains colons), which gets applied to the {} expression's input (remember, all filters have an input and an output).

{foo: .bar}

will produce the JSON object {"foo": 42} if given the JSON object {"bar":42, "baz":43} as its input. You can use this to select particular fields of an object: if the input is an object with "user", "title", "id", and "content" fields and you just want "user" and "title", you can write

{user: .user, title: .title}

Because that is so common, there's a shortcut syntax for it:

{user, title}

Not directly. You already know the name you want to use, so either hard code it:

jq -n --arg name value '{name: $name}'

or use a second variable

jq -n --arg key name --arg name value '{($key): $name}'

(I forgot about @0stone0's approach; that's probably what you want to use.)