Whether your site is a content site, an e-commerce portal, a museum site, or something else completely, you'll likely have fans. Fans come in two types: sneezers and non-sneezers. The word "sneezer" comes from Seth Godin's Purple Cow. A sneezer is someone who wants to tell others about your site or product. Usually because there is something remarkable to talk about.
Furthermore (from my brain now), sneezers like to have a way to spur that conversation. This is where merchandising comes in. By selling shirts, hats, shoes, and coffee mugs, you provide a way for others to get your product name out there.
You Won't get Rich from T-shirts
Unless you're George Lucas, you're not going to get rich from merchandising. Even if you come up with some hit meme, the margins you'll get from the t-shirts, hats, and thongs won't get you far. It's just nice to get a check now and then when you least expect it.
A few years ago, I tried selling products (Korean keyboards and keyboard stickers) and it just didn't pay off. The margins were too small to keep it up.
A good alternative is using Amazon aStores. The service allows you to set up a store at Amazon with your branding, with products you choose. The nice thing about it is that it takes care of itself. You don't need to worry about shipping products or handling returns, and you get a small affiliate amount from each sale.
For my aStore at zKorean, I set up some pages, each with keywords and categories of products that I want to display. Over time, Amazon will load in new products that match, meaning you don't need to go in regularly to add more. It's just automatically managed for you.
I'm not going to list all the things you can make at Zazzle, because it's way too many. But check it out. Yes, you can make Keds, but also posters, binders, postage, calendars, and so much more.
I use Cafepress for zKorean, and I did something a little different. Yes there are zKorean shirts and mugs, but also items for TaeKwonDo (many zKorean visitors are looking for TKD vocabulary). So even if your site isn't anything too awesome to sneeze about, perhaps you can find something in it worth merchandising.
Cafepress also has a wide array of cool stuff to slap your design on: keepsake boxes, beer steins, license plate frames, and more.
And the hoodies are warm. That's what I'm wearing on the About SND page.
An important point I must make is that your logo needs to be VERY large and high-quality if you want it to look good on printed matter. For more details, read this oldie.
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