How to change a product dropdown attribute to a multiselect in Magento

First, update the attribute input type to multiselect:

UPDATE eav_attribute SET
entity_type_id = '4',
attribute_model = NULL,
backend_model = 'eav/entity_attribute_backend_array',
backend_type = 'varchar',
backend_table = NULL,
frontend_model = NULL,
frontend_input = 'multiselect',
frontend_class = NULL
WHERE attribute_id = 'YOUR_ATTRIBUTE_ID_HERE';

Next, copy the attribute values from the old table to the new:

INSERT INTO catalog_product_entity_varchar ( entity_type_id, attribute_id, store_id, entity_id, value)
SELECT entity_type_id, attribute_id, store_id, entity_id, value
FROM catalog_product_entity_int
WHERE attribute_id = YOUR_ATTRIBUTE_ID_HERE;

Finally,  remove the old values or they will conflict with the new setup (the old values will load, but Magento will save new values to the varchar table):

DELETE FROM catalog_product_entity_int
WHERE entity_type_id = 4 and attribute_id = YOUR_ATTRIBUTE_ID_HERE;