drop index or constraint without knowing its name for Oracle

Your original version is mostly fine, except that you can't directly execute DDL in a PL/SQL block; rather, you have to wrap it in an EXECUTE IMMEDIATE:

execute immediate 'alter table MY_TABLE_NAME drop constraint "' || fName || '"';

This would be true even if the constraint-name were known at compile-time, but it's doubly true in your case, since fName isn't the constraint-name, but rather, a variable containing the constraint-name.

Also, this:

if (fName != '') THEN

is not valid/meaningful, since in Oracle '' means NULL. You should write

IF fName IS NOT NULL THEN

instead.

This is how to drop all constraints typed "R" for a column:

begin
    FOR rec IN (SELECT x.constraint_name fName FROM all_constraints x
        JOIN all_cons_columns c ON
        c.table_name = x.table_name AND c.constraint_name = x.constraint_name
        WHERE x.table_name = 'MY_TABLE_NAME' AND x.constraint_type = 'R' AND c.column_name ='MY_COLUMN_NAME')
    LOOP
        EXECUTE IMMEDIATE 'ALTER TABLE MY_TABLE_NAME DROP CONSTRAINT "' || rec.fName || '"';
    END LOOP;
end;