Here's the update code to include JustMarie's CASE method and Scott's CONVERT/CHAR(2) method.


DROP TABLE #TestTable
GO
DECLARE  @StartDate DATE = '2000' --Inclusive
        ,@LimitDate DATE = '2017' --Exclusive
;
DECLARE  @Days      INT  = DATEDIFF(dd,@StartDate,@LimitDate)
;
 SELECT TOP 1000000
        SomeDate = DATEADD(dd,ABS(CHECKSUM(NEWID())%@Days),@StartDate)
   INTO #TestTable
   FROM      sys.all_columns ac1
  CROSS JOIN sys.all_columns ac2
;
--===== PRECAST METHOD
  PRINT '========== PRECAST Method ==============================================================='
    SET STATISTICS TIME ON;
GO
DECLARE  @RunDay  CHAR(2);
 SELECT @RunDay = RIGHT('0' + CAST(DAY(SomeDate) AS VARCHAR(2)), 2)
   FROM #TestTable;
GO 3
    SET STATISTICS TIME OFF;
--===== DATENAME METHOD
  PRINT '========== DATENAME Method ==============================================================='
    SET STATISTICS TIME ON;
GO
DECLARE  @RunDay  CHAR(2);
 SELECT @RunDay = RIGHT('0'+DATENAME(dd,SomeDate),2)
   FROM #TestTable;
GO 3
    SET STATISTICS TIME OFF;
--===== DATENAME METHOD
  PRINT '========== MATH OVERRUN Method ==============================================================='
    SET STATISTICS TIME ON;
GO
DECLARE  @RunDay  CHAR(2);
 SELECT @RunDay = RIGHT(100+DAY(SomeDate),2)
   FROM #TestTable;
GO 3
    SET STATISTICS TIME OFF;
--===== CASE METHOD
  PRINT '========== CASE Method ==============================================================='
    SET STATISTICS TIME ON;
GO
DECLARE  @RunDay  CHAR(2);
 SELECT @RunDay =   CASE
                    WHEN day(SomeDate) < 10
                    THEN '0' + CAST(DAY(SomeDate) AS VARCHAR(2))
                    ELSE cast(day(SomeDate) AS VARCHAR(2))
                    END
   FROM #TestTable;
;

GO 3
    SET STATISTICS TIME OFF;
--===== CONVERT CHAR(2) METHOD
  PRINT '========== CONVERT CHAR(2) Method ==============================================================='
    SET STATISTICS TIME ON;
GO
DECLARE  @RunDay  CHAR(2);
 SELECT @RunDay = CONVERT(CHAR(2), SomeDate, 1)
   FROM #TestTable;
GO 3
    SET STATISTICS TIME OFF;

And, yes, I agree.... they all run in almost the same time.  I know we're making million row runs here so a 10 ms difference doesn't matter much but I am seeing the MATH OVERRUN method the Luis posted consistently coming in at about 10ms faster than the rest (about 70ms faster than the CASE method) Are you seeing roughly the same thing?


========== PRECAST Method ===============================================================
Beginning execution loop
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 343 ms,  elapsed time = 333 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 327 ms,  elapsed time = 331 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 328 ms,  elapsed time = 329 ms.
Batch execution completed 3 times.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.
========== DATENAME Method ===============================================================
Beginning execution loop
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 343 ms,  elapsed time = 336 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 328 ms,  elapsed time = 333 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 343 ms,  elapsed time = 334 ms.
Batch execution completed 3 times.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.
========== MATH OVERRUN Method ===============================================================
Beginning execution loop
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 312 ms,  elapsed time = 321 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 328 ms,  elapsed time = 320 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 327 ms,  elapsed time = 321 ms.
Batch execution completed 3 times.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.
========== CASE Method ===============================================================
Beginning execution loop
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 390 ms,  elapsed time = 389 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 375 ms,  elapsed time = 388 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 390 ms,  elapsed time = 390 ms.
Batch execution completed 3 times.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.
========== CONVERT CHAR(2) Method ===============================================================
Beginning execution loop
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 343 ms,  elapsed time = 344 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 343 ms,  elapsed time = 344 ms.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.

 SQL Server Execution Times:
   CPU time = 374 ms,  elapsed time = 374 ms.
Batch execution completed 3 times.
SQL Server parse and compile time:
   CPU time = 0 ms, elapsed time = 0 ms.