How to pass a table name for a cursor as function parameter?

Only values can be parameterized in SQL commands, not identifiers or key words. And it currently works only for SELECT, INSERT, UPDATE, and DELETE commands. Details:

• Error when setting n_distinct using a plpgsql variable

You need dynamic SQL with EXECUTE in plpgsql.
Combine that with the implicit cursor of a FOR loop, which is typically simpler and faster than an explicit cursor. Instructions in the manual.

I also suggest to use a regclass parameter to pass valid table names safely.

CREATE OR REPLACE FUNCTION myfunction(_tbl regclass) 
  RETURNS int  AS
$func$
DECLARE
   bid integer;  -- appropriate data type!
BEGIN
   FOR bid IN EXECUTE
      'SELECT id FROM ' || _tbl
   LOOP
      -- statements can use bid
      RAISE NOTICE '%', bid;
   END LOOP;

   RETURN 1;
END
$func$  LANGUAGE plpgsql;

Call:

SELECT myfunction('tableA');  -- careful: case sensitive!

More explanation:

• Table name as a PostgreSQL function parameter

And typically, there is a faster set-based solution that does not need to loop at all.

If you positively need a cursor, declare an unbound cursor and open it with OPEN FOR EXECUTE:

CREATE OR REPLACE FUNCTION myfunction(_tbl text)  -- could be regclass again
  RETURNS int AS
$func$
DECLARE
   _bids refcursor;
   _rec  record;
BEGIN
   OPEN _bids FOR EXECUTE 'SELECT id FROM ' || quote_ident(_tbl);  -- text must be escaped

   LOOP
      FETCH NEXT FROM _bids INTO _rec;
      EXIT WHEN _rec IS NULL;

      -- some logic
      RAISE NOTICE '%', _rec.id;
   END LOOP;

   RETURN 1;
END
$func$  LANGUAGE plpgsql;

Same call. More details in this closely related answer:

• Update record of a cursor where the table name is a parameter