Trigger function taking column names as parameters to modify the row
Problem
You had to pick the spot where all possible complications come together.

SQL (or PL/pgSQL) does not allow to parameterize identifiers. You need dynamic SQL with EXECUTE for that.

But the special plpgsql variable NEW in trigger functions is not visible inside dynamic code executed with EXECUTE.

And it's further complicated by passing column names as arguments to the trigger function in CREATE TRIGGER.

And it's not enough to make the target column dynamic, you want to fetch the source value from another dynamic column of the the row.


Unless you know your way around all the involved matters, I would rather try a simpler approach. Write a separate trigger function for each trigger and assign to the target column without dynamic SQL.
Solution
That said, it can be done with one line of code - and many lines of explanation. For the original example in the question, assuming this table definition:
CREATE TABLE tbl (
  tbl_id serial PRIMARY KEY,
  geom geometry,
  coords double precision[]
);

You need the additional module hstore installed (once per database) for this to work. Alternatively you could also use an undocumented feature of json_populate_record() / jsonb_populate_record() for the same purpose:

How to set value of composite variable field using dynamic SQL

CREATE OR REPLACE FUNCTION trg_demo()
  RETURNS trigger AS 
$func$
BEGIN
   EXECUTE format('SELECT ($1 #= hstore(%L, ST_SetSRID(ST_Point($1.%2$I[1], $1.%2$I[2]), 4326)::text)).*'
                , TG_ARGV[0], TG_ARGV[1])  -- target (geom), source (coords)
   USING  NEW
   INTO   NEW;
   RETURN NEW;
END
$func$ LANGUAGE plpgsql;
Trigger:
CREATE TRIGGER demo
BEFORE INSERT OR UPDATE ON tbl
FOR EACH ROW EXECUTE PROCEDURE trg_demo('geom', 'coords');

If you don't understand what's happening here, consider my advice above.
Explanation
I formatted the dynamic computation of the geometry bold to help you get a grip on this. Compare with the simple case below.
db&lt;&gt;fiddle here - with point instead of geometry, as PostGIS is not installed.
Old sqlfiddle.
Here is an even simpler version, just assigning the bare text value (which would require the target column geom to be type text). The simplified part in bold again:
   EXECUTE format('SELECT ($1 #= hstore(%L, $1.%I::text)).*'
                , TG_ARGV[0], TG_ARGV[1])  -- target (geom), source (coords)
   USING  NEW
   INTO   NEW;
The core feature is the hstore operator #= to (per documentation):

replace fields in record with matching values from hstore

Both target and source are columns of the new row, which complicates matters. If the source value was a constant we could simply:
NEW := NEW #= hstore(TG_ARGV[0], 'POINT(123.0 456.0)');

Related:

Assign to NEW by key in a Postgres trigger

But we need dynamic SQL to resolve the column name and fetch the source value.

Use format() to concatenate the query string safely.

TG_ARGV[0] and TG_ARGV[1] access the first two elements to the (0-based!) text array of arguments passed by CREATE TRIGGER.

%I concatenates the argument passed to format() as identifier (safe against SQL injection).

$1 references the value passed to EXECUTE in the USING clause.

We need the cast to ::text after calculating the geometry, because hstore takes text.

We need to decompose the row NEW for the assignment back, because plpgsql assigns rows as target column-by-column.