Is there a website that mimics Opera's Speed Dial Feature?

I haven’t seen one, but it wouldn’t be hard to put together.

I don’t think there is a software that mimics this to open programs or documents, but in Firefox, there are many add-ons (Speed Dial one of them, actually) with which you can take a clip of a certain part of a website or keep “smart bookmarks.”
If you’re using a Mac (and to go along with software to open programs/documents), there is a program called Quicksilver which uses just the keyboard in a series of shortcuts to open things.
Just suggestions, not sure if any of that actually answered your question.

The question is more about having a website that does this, so you can access it from anywhere. I keep a plain page on my site for this purpose, but it’s not fancy like the speed dial feature of Opera.

Firefox is working on a project called “Weave,” which saves stuff like bookmarks, etc. that can be accessed from any computer (presumedly with Firefox), but that’s the closest thing I can think of.

yep, still in beta though but looking very good: www.symbaloo.com