Why is the word "Honestly" in "Tukey's Honestly Significant Difference"? It seems unusual. Most statistical procedures are developed with the intent to give an "honest" answer. Why is that word used for this particular one? What's the historical background?
When doing multiple comparisons, the alpha value (Type I error rate) is accumulated over all tests. One method to do multiple comparisons is Fisher's Least Significant Difference (LSD).
Fisher's LSD computes the "least" or minimum value of difference required between two group means. If the difference between two group means is larger than this computed value, then the two groups are declared "significantly different".
However, LSD fails to maintain the overall Type I error rate. The true overall $\alpha$ value of the multiple comparisons test is not $\alpha$, but is inflated to $1-(1-\alpha)^c$, where $c$ is the number of pairwise difference tested. This can lead to incorrect conclusion.
Tukey's Honestly Significant Difference (HSD) is developed to combat this error rate inflation, hence the word "honestly significant", as compared to the error rate inflation being "dishonest".
Relevant read on LSD's weakness can be found at this question's answers.