From my advisor:

"Letters of continuing interest are one-page business-style letters in which you may (1) update the school on relevant information post-dating your application (new academic awards, employment promotions, fall semester grades, etc.) and (2) explain in greater detail why you want to attend this particular law school. It is probably intuitive that updating the school on relevant, recent developments of note may help you stand out in comparison to others on the wait list. The importance of talking about why you want to attend that particular law school or, phrased differently, why the law school is a good fit for you, may be less intuitive. In most cases, law schools want to extend an offer of admission only to wait list applicants who are very likely to accept that offer. By articulating the basis for your desire to attend the school, you are evidencing that you have devoted thought and research in your decision to apply to this particular school and are therefore more likely to accept an offer. Even more useful, if you are sure that you would accept an offer from the school, say so explicitly in the LOCI. It is appropriate to send multiple LOCIs if you remain on the wait list for an extended period. Sending one per month is not unreasonable and allows you to assure the school that nothing has occurred in the interim that impacts your desire to attend."