“It’s been 12 days since I last opened Tweetdeck,” said HubSpot CTO and co-founder Dharmesh Shah at the company’s Analyst Day today.

The morning’s agenda was a chance for reporters and analysts to see the company’s newly launched email marketing tool and to hear Shah and CEO Brian Halligan talk about the company’s future (more on that soon).

But those of us in the audience got an extra treat: a sneak peak at a new project Shah has been working on called SocialInbox.

“I want priority inbox for my Twitter stream,” said Shah, calling the process of trying to set up the perfect arrangement of columns in Tweetdeck a “poor man’s prioritization algorithm.”

The idea is to filter updates and mentions by their importance, including things like the credibility of the source (judged by SEO rank).

And, like Tweetdeck, SocialInbox isn’t just about Twitter. In fact, Shah’s goal is to create the single best 3rd party LinkedIn app in existence.

Shah showed the audience how SocialInbox could recommend 2nd degree connections on LinkedIn, and suggested that the system could eventually learn which of your contacts you actually know well enough to ask for an intro and then improve recommendations even further.

Shah claimed he wasn’t planning to show off SocialInbox until Halligan badgered him into it, and the company was careful to note that it is very much in its beta stages.

Still, the basic problem it solves is an interesting one. Right now Twitter doesn’t really have a backstop or a net to “catch” updates that fly by when I’m not using it. I still use Google Reader for that. But with the right filters, that could change.