1. Mathematical data about Bejeweled-type games. If I randomly fill an n x n grid with gems of k colors, how many moves will there be, on average? How many existing matches (i.e. gems that break before my first turn) will there be? How many new possible moves per turn does ideal play create, and how many does playing at random create (or remove)? And so on.
I could do the math on some of those questions with a little work (and I could Monte Carlo them even more easily), but not all of them, and anyway I wonder if somebody's already done it. Like that guy who proved there's no strategy for Tetris that guarantees you won't lose.
2. Match-3 games with strategic elements or substantial game mechanics outside the board. There's the wonderful-but-slightly-grindy Puzzle Quest (and a bunch of failed followups by the same company), and Dungeon Raid for iOS. Oh, and you could count Gyromancer, but its RPG elements seemed like window dressing to me-- the different creatures you summoned were interchangeable.
Maybe potential imitators of Puzzle Quest were put off by the fact that even the people who actually designed Puzzle Quest couldn't do it a second time. But seriously, look around the App Store or Kongregate-- the current state of casual match-3 development is the closest thing the gaming world has to an infinite number of monkeys. Something interesting must be happening.
By "match-3" I mean the Bejeweled mechanic but also "Bejeweled Twist" (rotate 2x2 sections of the board) and path-tracing games like Azkend or Dungeon Raid. Anything that involves matching tiles on a static board which refills itself whenever a hole appears-- as opposed to games where you fill the board yourself one piece at a time, like Tetris or Snood.