Hi,

I am working on building a simple list SPD workflow in SP2013 online site. I have site owner permissions. I have added "start task Process" activity and in the 'Outcome Options" section all the 3 dropdown lists are blank and I could not select any "task content type"

When I checked the list of content types in SPD, the list is blank. However, I could see all he content types in web browser, but not in SPD 2013. Any clue what is wrong with what I am doing? Do I need any specific permissions on he site apart from site owner?

You do need at least Read permissions in the top level site to access the list of Content Types. Are you using the same account for both SPD and the browser? Has the top level site blocked the use of SPD?