To normalize customer data in a staging data extension where a text field named 'birthday' contains date values in various formats, the developer can use the following SQL keywords and functions:

CASE, ISDATE, CONVERT (A) - These functions can be used to handle different date formats and convert them into a consistent format.

CASE, ISDATE, CAST (C) - Similar to the above, CAST can be used to convert text to date format if ISDATE returns true.

Example SQL query:

SELECT CASE WHEN ISDATE(birthday) = 1 THEN CONVERT(DATE, birthday) ELSE NULL END AS normalized_birthday FROM StagingDataExtension