One option, if you need it done in a pinch, and if your initial column was created with a default, is to create the new column you want, copy the contents over to it, and basically "abandon" the old column (it stays present, but you just don't use/update it, etc.)
ex:
alter table TABLE_NAME ADD COLUMN new_column_name TYPE NOT NULL DEFAULT '';
update TABLE_NAME set new_column_name = old_column_name;
update TABLE_NAME set old_column_name = ''; -- abandon old column, basically
This leaves behind a column (and if it was created with NOT NULL but without a default, then future inserts that ignore it might fail), but if it's just a throwaway table, the tradeoffs might be acceptable. Otherwise use one of the other answers mentioned here, or a different database that allows columns to be renamed.