The verb favourite or favorite (past tense favourited or favorited) is fairly new and isn't in many dictionaries. Two of the few are Oxford Dictionaries Online, who define it as:
record the address of (a website or other data) to enable quick access in future: you can see who else favourited the same pictures
favorite (third-person singular simple present favorites, present participle favoriting, simple past and past participle favorited)
1. Alternative form of favor.
1. (Internet) To bookmark.
1. (Internet) To add to one's list of favorites on a website that allows users to compile such lists.
Some people dislike the verb as ugly, others say it's "an abomination of a non-word" and "emblematic of cultural necrosis & linguistic stupidity", but it's commonly used by Flickr, YouTube and Twitter (with 75m, 800m and 200m users respectively) so it doesn't look like it's going to disappear any time soon. There's no entry (yet) at the Oxford English Dictionary, so when was this internet usage first used? Who popularised it and when?
I suspect it could have been first popularised by Flickr (launched 2004), then later boosted by YouTube (2005) and more recently Twitter (2006).
In other news, Flickr user fd has a new addition to his fantastic collection of Flickr toys— his random photo browser [...] fetches a selection of random photos that have been favorited by at least one person.
I found a verb use by Eric, a Flickr staff member, in December 2004:
But there are still a couple of bugs at play here: This photo should not appear in GustavoG's favorites, because he does not have permission to view it any more (it being totally private, and Gustavo not being a member of the fotolog group). I am guessing that he favorited it before it was made private. We will fix this.
(Interestingly, in the the bug report he replied to, GustavoG used the verb favored rather than favorited: "This was most definitely not a private page when I favored it.")
Edit: Stack Exchange also uses the verb favorite, and has done since at least 2010. If you click the star on this question then I may getf the Favorite Question badge for a "Question favorited by 25 users." (Or even the Stellar Question badge for a "Question favorited by 100 users.")