Can you add friends on the PSN website?

The ability to add friends via the website is a popular feature request (a quick Google search will turn up dozens of threads going back years asking for it), but it has never been added.

So as of now, you can only do it through the PlayStation 3, PlayStation Vita, or PlayStation 4 (oddly, you can't even add friends on the PSP).

Still can't add friends from the website, but you can use the official PlayStation iOS/Android app:

Very old question, but in case someone is googling this, now you can do it:

https://my.playstation.com/ and navigate to "Friends". From there you can now search, add friends, manage requests and everything else you could only do in the app before.