I just answered an email who asked how long it would take to get matches - since he had an uncommon surname. Here's my answer:
First - most researchers can become proactive - and try to find descendants of those families who are their possible genetic kin and convince them to test a family representative. Use paper trails, geographic proximity in earlier generations, and/or name spelling to target likely families. You can also test the most distantly known cousin that you can identify in your own family - which can confirm that your shared most recent common ancestor had the same yDNA as you.
Second - It really isn't a function of how common your name is - it's a function of how many genetic cousins you have and when one (or more) of them chooses to test.
If your ancestral family went through a population explosion (like the American frontier experienced) you may have many genetic cousins (like me) During the "frontier generations", my Barton ancestors were having 10-12 kids every generation - which creates a lot more opportunities to have kin with the same yDNA. So - if for whatever reason(s), your ancestral family wasn't having large numbers of children (or the sons weren't growing up and having their own sons) - you may not have as many genetic kin as I do.
Also - some families seem to have a tendency to have more sons. During those frontier years (and even in recent generations) my ancestral family was consistently having slightly more boys than girls - often 7 boys and 5 girls. This also helps generate more those genetic cousins - as your yDNA kin must descend from an unbroken line of males.
As example, of what can happen - both ways - here is my great-grandfather's situation:
My great-grandfather had 4 sons and 4 daughters. One son died as a child, another died at war and a third had no children. Only one son was left to carry on his yDNA. That son (my grandfather) had 3 sons and two daughters. His three sons combined to have 6 sons to carry on the family yDNA (and only 3 daughters) One of those 6 great-grandsons had only one daughter - so his branch has "daughtered out". Another of the great-grandsons had 3 sons and no daughters - but none of his 3 gg-grandsons has married yet. The other 4 great-grandsons each have one son of child bearing age who may carry on the yDNA. At this stage, there is only one ggg-grandson from the 7 gg-grandsons (who have an average age of about 30).
My great-grandfather's line isn't in immediate danger of "daughtering out", but it's looking like there won't be very many yDNA descendants. In this age of smaller families, my great-grandfather's line could be at risk in another one or two generations of "daughtering out".
So - be pro-active in finding and testing those families who may be your genetic kin. AND - if your family (or a branch of your ancestry) is down to only one male who carries the yDNA of your great-grandfather - be sure you test him while you can - to get his yDNA in storage. (25 yearsa t FTDNA)