I currently have a website in which I can add a feature that integrates with Twitter. I can add this as one of the features for a premium plan. Alternately, I can create a brand new website and have the Twitter integration feature be the the primary feature (I believe that this feature will have value by itself)
By adding to the existing website, I get the advantage of all the SEO work done and can leverage on the existing user base. However, if I build a new website, I may be able to get additional revenue.
I would suggest staying focused on adding value (premium is fine) to your core offering. Build where the traffic is.
Well its hard to pick what you are doing exactly without the website to understand the context.
Given the broader general question of I have a cool new idea growing out of an old one ... what should I do?
My answer to that again is it depends ... but if you have existing traffic I would most likely put it in the existing site first, use that as the incubator for the concept and then if it looked like a winner extract it out later. If you design with this end goal in mind you should be alright.
This mainly comes down to the trial and error approach for these new ideas, its generally harder to get a good concept incubation and feedback loop set up than it is to write the new bit of code.
... integrate into #1, also make #2, run some tests, different look and feel and marketing angle (not too different) and get some more feedback. Why guess when you can know for a low cost.