I've implemented two algorithms. 1) look which letter(s) appear in the most words which are still possible, pick the one which appears in most words. Break ties randomly. 2) look at all the letters that appear in the words that are still possible, pick one weighted on frequency (so, if there are 15 words left, and 'l' appears in 10, 'm' in 5, than the chance of picking 'l' is twice the chance of picking 'm').

Using this with a dictionary of 355k words, algorithm 1) always find 'rhythm' with no guesses to spare (7 failures). algorithm 2) sometimes fails to find 'rhythm' in 7 failures or less, but the average number of failures is less (6.60). This surprised me.