Tweet value prediction: What sort of analysis (Bayesian?) might predict how much a Twitter user will value a tweet?

In general you'd probably need some supervision for that, and a user marking 10 or so tweets as high-value is far too little supervision (this is not enough for you to zero in on specific words, for example, or to find out the weights for the people followed by your user). Luckily, this is essentially the same problem behind the gmail priority inbox. There was a paper on a recent workshop describing their approach in more detail, but the essential ideas are:

1. Define a metric of what is a valuable tweet for your users. One simple way of doing this is saying that a tweet is valuable if links on it are clicked, it is retweeted, or it is replied in a certain timeframe after being posted.
2. Learn a classifier that tries to predict wether or not a tweet will fit those criteria. This is a natural setting for online learning algorithms, since first the algorithm has to predict a relevance for a tweet and soon afterwards it is told wether it got that right or wrong (and this is why the timeframe above is imporant). The google people use an online logistic regression algorithm that has the nice bonus of giving you well-calibrated probabilities that you can tune. Another nice feature of online algorithms for this problem is that they are naturally suited to forgetting past behavior and adapting to changing preferences (this is achieved by using a too-aggressive learning rate schedule)
3. Use as many interesting features as you can in that classifier. A nice thing of the google approach is that they learn both a global classifier for all users and a local classifier for each user that adapts to his/her preferences. Hence a new user will already have some pretty good defaults as soon as he installs your application (for example, @replies are usually important, tweets from people you're following recently are also more easily important, as are tweets of people with many followers or many retweets).

I recommend you read that paper, as it also explains details of their algorithm, environment, and assumptions.

How about the following idea for modeling user preferences. For all tweets in a user's history you compute tf-idf (or only term frequencies) feature vectors. You weight each of these tf-idf vector by some quality score which should be computed as suggested in Alexandre's answer, say #click/#impressions also time to make the latest tweets more important. Then sum them all up, therefore you are feature vectors won't be as sparse as the feature vectors for the tweets themselves. Then you can use LDA, LSI or some unsupervised clustering, to get better and semantic representations for these pseudo-documents which model user preferences. Next time you want to rank new tweets for a user, compute their tf-idf, map them in the latent/semantic space and rank them according to euclidean distance to the pseudo-document for this particular user. (there are some details to be worked out but generally I think it should work out.)

This is kind of a simplistic approach but:

Try to run LDA over your current tweets, so you know which topics is the user more interested in, although, I'm not sure how LDA would work with 160 characters, you can always concatenate tweets.

Then, with that, try to run an online LDA to analyze incoming tweets, and with that you might get an idea on how interesting is the tweet for him.

You can go crazy and try to use hirerchichal models to look for meta topics relating all the tweets of the user.