In "things I think someone should make":
An app called When?, with a big button labeled Now. It records the dates and times that you press the button.
There's a smaller button that takes you to a screen where you can see the gathered data analyzed in various ways. Do you press the button at certain times of day? Certain days of the week? Days of the month? Times of year? Phases of the moon? Or you can view the data noncyclically -- have you been pressing it more or less often over the whole course of the time you've been using the app, or maybe there was a temporary surge or gap six months ago, or something.
You use it as a super-simple lifelogger. Pick any category of thing, and you can track when it happens. When does it rain? When do you get headaches? When do you eat ice cream? When do you get hot flashes? You can use When? to answer any of these questions.
Probably there should be a way to track more than one thing, but that would make the interface less elegant.
With a desktop app, you could create a bunch of different aliases (shortcuts, symlinks, whatever they're are calling it these days), and detect how it was invoked with $0. (Or whatever the equivalent is on non-Unix-like OSes.) But on a mobile platform, the doctrine of One App, One Icon seems pretty ingrained. This seems like an unfortunate limitation.
(This is why we need Free Software.)
Maybe there could be a sister app, called What?, that has a list of things and you tap the one that happened. It's not quite as elegant, but it gets the job done.