Which of the following is correct and why?
- opportunity at the tips of her fingers
- opportunity at the tip of her fingers
The former seems correct to me, but I don’t entirely understand why.
Tips is correct because fingers is plural. Since there are several fingers, there are several fingertips. Other examples:
On the other hand,
There's only one tongue, so of course there's only one tip.
As I have always done, I will use my own language to help in this answer. Supposedly, when you say "cars", you mean more than one. When you talk about "the motor of the cars", you're referring to a unique model of motor that works in every car, so the motor is repeated after every car, being the car different, but not the motor.
After that, you could say "cars" and then talk about the "the motors of the cars" and you'll be referring to every motor of every car but never stating that they are the same. The previous sentence implies that the motor is the same for the cars.
In your case, the fingers have tips. But the fingers have everyone, a tip. And that's the clue. You may use both with the only difference that the singular form is a generic or general to denominate repeating objects that are almost the same, and the plural, that reveals that being repeating objects almost the same doesn't make them equals.
Sometimes, when you're being so specific, it just has real importance when you're using it (the sentence) in a phrase that requires a clear definition about the question of being many or just one part that repeat in many others. If that is not the case (an expression for example) I don't think you need so much clarity.
Giving a good look at the other answers I realized they are right about the metaphorical use of tip v/s tips. I think that's also a good point.
Personally, I'm quite happy with He spoke without any notes but seemed to have every fact at the tip of his fingers, but I find opportunity at the tips of her fingers a rather odd metaphoric usage.
Checking Google Books, I find 28 hits for both plural "at the tips of his fingers", 11 for "mixed" "at the tip of his fingers", and 8 for both singular "at the tip of his finger" (obviously "at the tip of his fingers" is nonsense). Note that most of the results seem to be completely literal anyway (Andre looked at the tip of his finger).
For me, the "standard" metaphoric usage is "at his fingertips". I'm not going to try and count the relevant hits in that link - but it's at least tens, if not hundreds of thousands.
Having said all that, regarding OP's specific usage, I think the "correct" version would be "at the tip of her fingers", since the intended sense is of a singular metaphoric reach, ability to access easily.
But if, say, an investigative journalist were talking to his editor, he might more correctly say "I have several reliable sources at the tips of my fingers", since in that case he'd be talking about easily reaching different people in different places.
When you say "the tips of my fingers", you are referring to the individual tips of each finger. A possible usage would be "I felt a tingling in the tips of my fingers".
However, when you're talking about a more metaphorical something that's just within reach, you would use "the tip of my fingers", since you're referring to all your fingers - your figurative "reach" - as one unit.