So your standard happy birthday in Japanese is お誕生日おめでとうございます。My question is, what sorts of additional nice things can one say on someone's birthday. Like, in English you might say something 'Hope you enjoy your special day' etc.

素敵な誕生日を過ごしてください。is a pretty close equivalent for your example.

Googling the above I found a page with the reverse of your needs, maybe it will help?
http://mysapu.com/message/birthday_message/id00184/
http://mysapu.com/message/birthday_message/id00075/