Many amateur sites have a "You are visitor number X" counters on them. What effect does this button have on visitors to the site? Does it increase their trust in the site?
It's social proof, of a sort... they want to show how many people are using the site, so you feel compelled to use it as well.
"Well, if 238 other people have visited this site, how could it possibly be wrong?"
However, the key word here is "amateur". Simple visit counters have no place on modern webpages. Instead, most modern sites rely on more refined social proof techniques, such as "popular" content listings, likes & tweets, view counts for individual pieces of content, comment counts, etc.
I think there's a good reason these have fallen out of favor due to the "liked"/"shared"/ect style social media plug ins. Raw visit count isn't that meaningful especially in regards to quality. But if you can explain that people liked or shared content that's much more effective at expressing value, especially when you're showing a trusted metric. People trust the Facebook/Twitter like/share counts; those are the developers' APIs.
Who knows that that slowly upticking visit counter is reading? Even if accurate they just don't look as appealing or trustworthy in my experience. I have personally met a few people that expressed complete distrust in such visit counters, assuming they're faked. Not real research but something to consider.
Activity indicators, such as visitor count (or recent news items, or for example 'date last edited') show that the site is alive and that encourages trust. Showing signs of other people on the site is social proof (this includes visitor counts, like buttons, customer reviews, etc.). In many cases the number of visits isn't the best metric to show. (But note that here on this site the questions the number of visitors for each question, but named the number of views.)
Additionally, for some amateur sites, the visitor count button may be of more value to the owner (and his family/friends) than to the visitors of the site. If you propose to the site owners to remove their visitor counts, they don't say that it's important for their visitors, but instead tell that they themselves really like it to keep an eye on the activity on their site. In that case, a few simple pointers to other analytics tools may suffice.