Check If Record Was Inserted Or Updated In Table
As many have commented, you want to isolate your trigger to fire and modify only when necessary.  This means your going to have to filter on just the record that was inserted.  You can do this with the inserted view.  As you will see from the sample code below, I've slightly modified your code to more easily see what's going on and isolate the update statement to only affect the inserted row.

--create the table
create table TwoNumericNames (rowID int IDENTITY, name nvarchar(500))

--insert a row
INSERT TwoNumericNames
VALUES (N'AA')

--note the row
select * from TwoNumericNames;
go

--create trigger
CREATE TRIGGER trgUpdateName
ON TwoNumericNames
FOR INSERT
AS
Begin

    UPDATE T
    SET T.[name] = CASE
                    WHEN I.[name] LIKE '%AA%' THEN 'Alpha Awe'
                    WHEN I.[name] LIKE '%RR%' THEN 'Raw Red'
                    WHEN I.[name] LIKE '%CC%' THEN 'Capt Cree'
                    WHEN I.[name] LIKE '%MM%' THEN 'My More'
                    WHEN I.[name] LIKE '%EE%' THEN 'Ed Ewe'
                    ELSE I.[name]
                END 
    FROM TwoNumericNames T
    JOIN inserted I ON I.rowID = T.rowID

END
go

--insert another row
INSERT TwoNumericNames
VALUES (N'RR')

--note only the one record is updated
select * from TwoNumericNames