com.google.android.googlequicksearchbox is the name of the GOogle Search app.
I see it starting around 27th April, from Android devices, naturally. From the actual referrer, my guess would be a link from the default Google search bar on a device (this opens in the Google app rather than in a browser). This is installed by default on most devices, I believe.
According to the Play Store [play.google.com], the app updated on April 28th, too.
Hmmm I notice this referrer a number of times as well this past week. I was wondering what it was. My first thoughts was Google Play but as I don't have any apps for my site I figured that wasn't it. Andy's analysis makes the most sense.
I haven't had a chance to test this yet, but I'm hoping I'm correct. "Direct" as a visitor source in a typical analytics package seems to include an increasing amount of traffic from other sources, these days - this would at least help with a proportion of mobile web searches.
I did test this, and it is from the Google search app installed by default on Android devices.
If you use Google Analytics, it will show up as referral traffic - it should probably be listed in organic search. Ideally, it could be added via tracking info >> Organic Search Sources however, that demands a query parameter which is not sent with this traffic. I guess it could be added with dummy parameter, but I don't know if this would track or not.
I am also seeing "googleweblight.com" being reported by Google Analytics where "android-app://com.google.android.googlequicksearchbox" never came up in Google Analytics just my own internal statistics. I haven't seen googleweblight in near the numbers as the other one which makes sense since it is supposedly from other countries which has slow internet.
Googleweblight is a proxy that "intercepts" the search result click if Google thinks your site is too slow - so, should only appear for slow connections and if Google thinks your page will be slow. I've seen that since about October last year in small numbers (mostly from India).
"googleweblight.com" being reported by Google Analytics where "android-app://com.google.android.googlequicksearchbox" never came up
googleweblight is the User-Agent. android.quickblahblah is the referer. You wouldn't expect to see them in the same places.
* I know of one humanoid visitor who routinely does this.