A new widget showing the days within the selected period (or the 24 hours if the period is "day"), should be easy to build and sounds like an important widget to provide in the dashboard.

This feature is often requested, another one: http://forum.piwik.org/read.php?15,120427

In this forum post I think for the first time someone requested to have this similar feature but for Email reports Graphs.

Looks like a significant number of issues have been marked as duplicates of this one.

Maybe this suggests that the UI of the timeframe selection isn't the clearest possible?

@dandv indeed, we should and will fix it, not sure when it will be possible as our resources are limited.

Here is example of this feature in a Google Product:

thanks for the screenshot from google! Looking at it there is a small but important difference: google talks about "daily" and piwik talks about a "day"+ we select a date - so the user has completely different expectations...

for piwik:
selecting a day and showing data for only this day should be really consistent in large graph, in small graph previews and in all tables

doing it any other way is a huge problem (as also reported in #8917 and in several other ticket):
the user select something and got the results for something else

on other places e.g. in visitor log the user gets what he has selected (the place I'm often working with)
So how should one notice and understand this inconsistency? (sorry for these hard words)

visits by hour (when period is day) (eg. requested in #7498)

I don't think this is possible. We don't have hourly data for all graphs.