I looked in the "friends" section of the PSN website, and I couldn't find any way to search for other friends to add.
Can this be done anywhere online or can it be only done through the console system?
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 or PlayStation Vita (oddly, you can't even add friends on the PSP).
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?