Is there a way to make a TSQL variable constant?

Is there a way to make a TSQL variable constant?

No, but you can create a function and hardcode it in there and use that.

Here is an example:

CREATE FUNCTION fnConstant()
RETURNS INT
AS
BEGIN
    RETURN 2
END
GO

SELECT dbo.fnConstant()

One solution, offered by Jared Ko is to use pseudo-constants.

As explained in SQL Server: Variables, Parameters or Literals? Or… Constants?:

Pseudo-Constants are not variables or parameters. Instead, they're simply views with one row, and enough columns to support your constants. With these simple rules, the SQL Engine completely ignores the value of the view but still builds an execution plan based on its value. The execution plan doesn't even show a join to the view!

Create like this:

CREATE SCHEMA ShipMethod
GO
-- Each view can only have one row.
-- Create one column for each desired constant.
-- Each column is restricted to a single value.
CREATE VIEW ShipMethod.ShipMethodID AS
SELECT CAST(1 AS INT) AS [XRQ - TRUCK GROUND]
      ,CAST(2 AS INT) AS [ZY - EXPRESS]
      ,CAST(3 AS INT) AS [OVERSEAS - DELUXE]
      ,CAST(4 AS INT) AS [OVERNIGHT J-FAST]
      ,CAST(5 AS INT) AS [CARGO TRANSPORT 5]

Then use like this:

SELECT h.*
FROM Sales.SalesOrderHeader h
JOIN ShipMethod.ShipMethodID const
    ON h.ShipMethodID = const.[OVERNIGHT J-FAST]

Or like this:

SELECT h.*
FROM Sales.SalesOrderHeader h
WHERE h.ShipMethodID = (SELECT TOP 1 [OVERNIGHT J-FAST] FROM ShipMethod.ShipMethodID)

My workaround to missing constans is to give hints about the value to the optimizer.

DECLARE @Constant INT = 123;

SELECT * 
FROM [some_relation] 
WHERE [some_attribute] = @Constant
OPTION( OPTIMIZE FOR (@Constant = 123))

This tells the query compiler to treat the variable as if it was a constant when creating the execution plan. The down side is that you have to define the value twice.

No, but good old naming conventions should be used.

declare @MY_VALUE as int

There is no built-in support for constants in T-SQL. You could use SQLMenace's approach to simulate it (though you can never be sure whether someone else has overwritten the function to return something else…), or possibly write a table containing constants, as suggested over here. Perhaps write a trigger that rolls back any changes to the ConstantValue column?