It appears that the reason this has been a problem for so many is that it works differently than it used to. As noted in the original post, in Word 2010, all you had to do was select Format > Highlight, and it worked.
Now in the first step you have to explicitly select a highlight color. This is actually not a bad thing, because (in my experience) in Word 2010, yellow was the only color available.
The detailed steps are:
Go to Home > Font group and select a highlight color.
Go to Home > Editing group > Replace.
In the Find What box, type the text you want to find and highlight.
In the Replace With box, type ^& (I.e., instruct Word to replace using the same text string as in the Find box. You could also just leave the Replace box blank.)
Click the Format button and select Highlight. ("Highlight" appears below the Replace With box).
Replace the text with highlighting by stepping through it with Find Next/Replace or by clicking Replace All.