"As early as" is not, as your tagging originally suggested, an idiom.
"Early" essentially denotes the beginning of a section of time. That's not a particularly technical way of reading it, but it will work for our purposes.
Following that, "as early as" indicates the beginning of a time opening. Your first given example could be rephrased thusly.
The money could be paid any time at the beginning of next week or later.
In your later example, "I wake up as early as 6 o'clock," you could rephrase that as something along the lines of this.
The earliest I wake up is 6 o'clock.
As you might notice, however, both are open-ended. Following the first example, I could pay at the beginning of next week, or I could wait three months. There's no ending to what it says, the boundary specifies only a minimum. The same is true in the latter example: I could wake up at six days of the week and 6:00 am one, and still say "I wake up as early as 6:00 am."
Your middle example isn't an "as early as," but "early" still keeps the same meaning. "Early booking is essential" simply means that booking near the beginning of your ability to do so is necessary. If the service begins offering tickets at noon, you'd better book as soon as possible after that starts, or else you might not be able to.
This is similar in open-endedness to the previous examples, insofar as it is also open-ended. This says "book early," but doesn't outright state any ending parameter for when tickets will no longer be available.
It occurs to me that I failed to recognize the nuance of usage here. In certain cases, a speaker is unlikely to express "as early as" to actually mean an open-ended remark. A statement like this is meant generally as more of an empty promise. It's similar to "up to" in that sense. So at that point, you have to look to context, and the motives of the speaker.
Some advertisements on television might say you'll receive your shipment "as early as the following week." But that could mean only if you pay for express shipping, and that regular shipping would be several weeks. On the other hand, if someone is making a promise they can't realistically back out of (as, Maciej kindly pointed out), "as early as" will probably refer to an estimated completion time, because the estimation being inaccurate wouldn't be good for the person who originally said it. "As early as Monday" would probably mean "Monday or Tuesday" in this case.
As with many other aspects of language, unfortunately, it depends a fair bit on context, and what you think the person stating it is trying to convey. I stand by the technical aspects of what I said in the beginning, that it's deliberately open-ended. But that said, Maciej was right to call me out, because I didn't include its practical usage.
In essence, one can take an "as early as" statement as a suggestion of a point in time, and the validity of that suggestion should be assessed on a case-by-case basis, with emphasis placed on the consequences of the suggestion being inaccurate.