Using source columns in OUTPUT INTO clause of an INSERT statement (SQL Server)

Actually, you can achieve the same thing by changing your INSERT to a MERGE. While the MERGE statement is actually a pretty neat way to do "upserts" in SQL Server, there's nothing to stop you from using it just for the purpose of inserting:

-- The existing table 
DECLARE @MyTable TABLE (ID INT IDENTITY(1,1), [Name] NVARCHAR(MAX));

-- My data I want to insert
DECLARE @MyInsertData TABLE (ID INT, [Name] NVARCHAR(MAX));
INSERT INTO @MyInsertData ( ID,Name)
VALUES ( -1 , 'bla'),(-2,'test'),(-3,'last');

DECLARE @MyCrossRef TABLE ([NewId] INT, OldId INT);

MERGE INTO @MyTable AS dest
USING @MyInsertData AS ins ON 1=0   -- always false

WHEN NOT MATCHED BY TARGET          -- happens for every row, because 1 is never 0
    THEN INSERT ([Name])
         VALUES (ins.[NAME])

OUTPUT inserted.ID, ins.ID
INTO @MyCrossRef (NewId, OldId);

-- Check the result
SELECT * FROM @MyCrossRef

One of the nice things about MERGE is that it allows you to access the source columns as well as the built-in inserted and deleted tables in the OUTPUT clause.

My code may contain errors, as I haven't actually tested it. My blog post from a few years ago goes into a little more detail, including on query performance.

The output clause can only access data in the target rows and constants/variables, not data from elsewhere in the source SELECT, like if you were operating in a trigger.

https://docs.microsoft.com/en-us/sql/t-sql/queries/output-clause-transact-sql states:

Any reference to columns in the table being modified must be qualified with the INSERTED or DELETED prefix.

So to get the original ID you would need to include it in the destination table so that the output clause can echo it back, like so:

-- The existing table 
DECLARE @MyTable TABLE (ID INT IDENTITY(1,1), [Name] NVARCHAR(MAX), SourceID INT);

-- My data I want to insert
DECLARE @MyInsertData TABLE (ID INT, [Name] NVARCHAR(MAX));
INSERT INTO @MyInsertData ( ID,Name)
VALUES ( -1 , 'bla'),(-2,'test'),(-3,'last');

DECLARE @MyCrossRef TABLE ([NewId] INT, OldId INT);

INSERT INTO @MyTable ( [Name], SourceID )
   OUTPUT Inserted.ID, Inserted.SourceID INTO @MyCrossRef
   SELECT [NAME], ID FROM @MyInsertData INS

-- Check the result
SELECT * FROM @MyCrossRef

though altering the schema of the target object may not be practical in your situation so this may not be applicable.