If the project is called 'Chat Widget' it's ok.

"Goodbye" is the common, ordinary, standard word. "Farewell" is old-fashioned. It is formal and dignified. It is sometimes used in a slightly joking way, because you are using nineteenth-century language in the twenty-first century. "Farewell, Chat Widget" sounds informal and jokey. It might be a good choice if your news story is written in a friendly, conversational tone. I would put a comma after the word "Farewell."

A slight difference, depending on what you want to express. Farewell, you are parting with someone or something and it is like a well wishing occasion, party, speech, expression of or a gesture of appreciation. You can say or organise a farewell party, farewell speech… Whilst goodbye doesn’t convey any well wishing message.