Declare a variable in RedShift

Actually, you can simulate a variable using a temporarily table, create one, set data and you are good to go.

Something like this:

CREATE TEMP TABLE tmp_variables AS SELECT 
   '2015-01-01'::DATE AS StartDate, 
   'some string'      AS some_value,
   5556::BIGINT       AS some_id;

SELECT *
FROM Orders
WHERE OrderDate >= (SELECT StartDate FROM tmp_variables);

The temp table will be deleted after the transaction execution.
Temp tables are bound per session (connect), therefor cannot be shared across sessions.

Slavik Meltser's answer is great. As a variation on this theme, you can also use a WITH construct:

WITH tmp_variables AS (
SELECT 
   '2015-01-01'::DATE AS StartDate, 
   'some string'      AS some_value,
   5556::BIGINT       AS some_id
)

SELECT *
FROM Orders
WHERE OrderDate >= (SELECT StartDate FROM tmp_variables);