Jeez, I'm so late to the party that all the good answers are taken. :-p I'll only expound on what's already been said, or add new perspectives.
I think the most important aspects to work backwards from are A) the brand at play, B) the type of website it is (if business, what vertical), and C) its users. Get ready for Supremely Extreme Generalizations™!
Brand and website type should drive your broad palette. Brand is the underlying emotional core of your sounds, the equivalent of the driving themes in a film or the emotional content of a scene: Warmth, musical vs. texture, etc. Business will shape the texture of your sounds: High tech, organic, bubbly, severe, minimal, rich, etc. As others have said, this is where discussions with clients is key. Get emotional brand descriptors, those are critical.
Users are what gives your sound context. A website whose features will be consumed by busy people during the day will probably warrant a different approach than a website meant to be used at home, at night. Part of the challenge of online UI sound design is the same as designing for TV: You have no control over the end playback device, or the room or even the context surrounding how people are consuming the media. I also seemed to have coined the term "least common expectation," the emotional equivalent of the "least common denominator." Like visual effects, the kind of user you have will influence how much they're even expecting sound, and if they are, what their aesthetic minimum expectations are. And, people being people, they are the ones triggering the sounds, so like in game sound design, you need to 1,000% ensure that your sounds won't trigger fatigue...having slight variations triggered with some very simple programming can pay off bigtime. Please, treat button presses like foley footsteps in games, your users will LOVE YOU for it.
Over the last 3 years, I've found most of my work in this area breaking down into two types: Musical effects and digital effects for interfaces. Vast variances in each, but very distinctive. Listen to the XBox menu system for a great example of musical UI effects; if you like abstract IDM or "microsound" electronic music, then those ultra-short, crisp, digital sounds can often work well in interfaces, too, on the other end of the spectrum.
Finally: Be consistent. Rollover, mouseDown, mouseUp...be consistent in what user input triggers sounds. The richness of the visuals often guide your hand and ears here. Rollover and mouseUps are minimums, in my opinion, but don't forget about screen transitions, lists unfurling, alert messages...UI is about a LOT more than buttons.
My best recommendation is really to think about every digital UI sound design project as a device or product: Whole unto itself. It'll get your work past the "sound for website" stage and into the "digital interface effects" stage. My Roomba, my Mac, my PC, and my MINI Cooper have completely different sonic vocabularies, and it really does imbue each of them with different personalities.
Sorry I'm late, hope that helps!