UPDATE my_table
SET name =
CASE
WHEN Original = 'a' THEN 'x'
WHEN Original = 'b' THEN 'y'
...
END
That will update EVERY row. So if there's an Original value you haven't specified, it will be set to NULL. So you might want to limit the update to just those you want to update, with a WHERE clause, like so:
WHERE Original IN ('a', 'b', ...)
OR, as an alternative, you could use an ELSE statement, which leaves the name value as is, if it doesn't have a match in the WHEN statements, like so:
CASE
WHEN Original = 'a' THEN 'x'
WHEN Original = 'b' THEN 'y'
...
ELSE name
END