Publicity teams would absolutely pay for a "Google Alerts but Way Better" product. You'd need to make it much better and easier to use, and have features similar to the peak tracking so that they can get a sense of how popular a certain word is.
They would probably also like to have tweets and blogs from accounts with higher follower counts, pagerank, etc prioritized. So if you get a tweet from an industry leader, you get a text immediately. If you get one from a new account with 4 followers, it gets folded into the daily report as a statistic.
Your point of differentiation would be a "secret sauce" of presenting only the most important information to users and not bothering them with all of the irrelevant ones.
Publicity teams are generally quite free with the company card, so I'd suggest a freemium offering but with a very limited basic plan. And you don't need to worry about them creating multiple accounts because publicity folks don't have the time or knowhow for that.
edit: Drop me a line if you're building something that would work for non-tech savvy publicity folks and need beta users.
