Sry for the long response time, I had to leave for class shortly after I posted that. Pretty interesting question. I think that the obvious would include general activity like comments, replies, and the vote system should be incorporated, but how.
DEFINITIONS
So, I'm going to define relevant as it relates to the topic stated, as otherwise I just don't see anything else you could really measure for relevance with the current system. There is no tag system, and the current community system already has problems staying true to the community (just check Site Feedback, filled with Server Status complaints, balance things from pbe still occasionally happen, and also some things that better fit Help & Support) Also, considering that what may be relevant for you might not matter to me, I'm going to restrict "relevant" to meaning "relevant to the topic, AKA staying on topic".
I'm also going to define "fresh" as new activity. This means that the more recent the actions taken are, the more weight that particular factor is in deciding the ordering on the front page.
OPINION
How I would achieve this would weigh total vote count minimally, and direct replies to the OP much higher, with subsequent response levels weigh a little lower with each layer. If a post is simply being upvoted, while it will still help, if there is no discussion going on, then why is it on page 1? Likewise, if all the posts are a response to the top comment, then chances are a derailment is going on. That said, if there is alot of traffic in that first comment, and the other factors are weighing well, the discussion could still hit the first page.
Also, I would take a total upvote/downvote ratio of the thread, and the closer to a range around 1 - 1.5 ratio (this number can be played with) means that the thread has more "relevant" material, not just circlejerking/counter-circlejerking. In other words, if the thread is mostly upvotes, or mostly downvotes, the less likely it is to be seen. The idea is that if all that is happening is everyone agreeing and upvoting, then chances are that there isn't a whole lot to discuss, while on the flip side if everyone is downvoted hard, then chances are the thread isn't really being productive.
Lastly, the Rate at which actions are happening in the thread would be a large factor. Frankly, if alot of discussion is happening, that is worthy of recognition.
To prevent Bumping, or any anomilies: these factors will ignore the auto self vote that happens when you post, and actions that happen sequentially by the same person (like upvoting then responding will only count as 1 action, just like five posts replying to yourself will essentially be 1 action).
This weight will degrade over time if it isn't updated. In otherwords, A mega thread can stay near the top for a while, but if the discussion dies down, it will eventually drift downwards.
I hope that was readable, and it made sense. Feel free to ask me to clarify on any points you didn't quite understand. As a footnote, I would like to know if you have any plans to help out lengthier posts like this one. Thanks for making it through!