Why is it important to protect speech, even if that speech is unpopular?

Speech that is unpopular today could be widely held tomorrow. Once upon a time, the idea that burning witches was wrong was deeply unpopular. Yet nowadays, thank goodness, just about everyone agrees with it. In the marketplace of ideas, the good, the bad, the true and the false, the popular and the unpopular, should be allowed to fight it out among themselves without being suppressed. This way, so the argument runs, the truth will out.
Even if this doesn't always happen, a moment's reflection will tell us that free speech is generally the best way to go. Just because an opinion is unpopular doesn't necessarily make it untrue, and vice versa. There might be all kinds of reasons why a particular opinion is unpopular, reasons that have little or nothing to do with its truth-value.
That being the case, the best approach is to give each and every opinion, no matter how unpopular, the chance to be examined, scrutinized, and criticized. Under those circumstances, what invariably happens is that we are then able to find out why such opinions are unpopular, and if so, whether they ought to be. Having examined them, we may well continue to reject them, but at least then we'll be doing so from a position of much greater understanding, and for that we require free speech.

Much of the analysis of the role of free speech in democracy derives from the seminal work On Liberty by John Stuart Mill. Mill argues that truth is discovered in free debate, with ideas tested by having a vibrant public sphere in which they are expressed openly. Incorrect ideas can be disproven and valuable ones circulated and confirmed.
Many new ideas are unpopular. The notion that the earth was spherical rather than flat and that it revolved around the sun were both unpopular, and the germ theory of disease was considered absurd. Many scientific innovations that are now universally accepted started as unpopular innovations. Had these been permanently suppressed, many important scientific and medical advances would not have happened. Just because ideas are popular does not mean that they are true, and in that same vein, unpopular ideas are not necessarily false.
Finally, suppression of unpopular ideas (or people, or groups) is a step towards authoritarianism, preventing challenges to what is sometimes called "the tyranny of the majority". Many contemporary advances such as racial equality, gender equality and LBGTQ rights started off as unpopular ideas.