That search is a difficult one, as just about every feature users might imagine in a smartphone -- web browser, music player, GPS, camera, flashlight -- is already here as a fully mature technology.
At the recent Mobile World Congress in Barcelona, Spain, several companies were showing prototype of phones they hope might do the trick -- phones with dual screens.
There have been such phones before, but they have so far failed to capture the public's imagination, either because the technology has been less than perfect or they've been seen as just too "odd" as a mainstream offering.
One company bucking that trend and hope that "more is better" is Japan's NEC, which is set to introduce a new smartphone in Japan in April, the Medias W, which will feature a second screen that can fold out to double the available screen area.
The phone features 4.3-inch screens front and back, but the back screen can be folded around to sit beside the front one, effectively doubling screen real estate.
Users can treat the doubled screen as a single display -- if they can deal with the narrow black band down the center where the two screens butt each other -- or each screen can run separate applications.
Running two apps simultaneously is at the moment something no single-screen Android phone can do.
NEC may not be alone in eventually offering such a phone to the marketplace. Blackberry has filed for a patent for its own dual-screen concept, which can also run separate apps.
Patent applications often -- even usually -- don't make it as far an actual devices offered for sale, but Blackberry might just consider it as something to help it claw its way back as a competitor in the smartphone arena.
Russian phone maker Yota was also at Barcelona with a phone that takes a different tack when it comes to dual screens. Their smartphone features a conventional color display on the front, while the back of the phone features a monochrome e-ink display.
The e-ink screen can be used like a traditional e-reader for books and magazines -- although it's a bit small for extended reading sessions -- and it can also display text messages and other alerts without having to wake the phone, thus putting less drain on the battery.
Both the color LCD screen and the e-ink screen are 4.3 inches, becoming something of a standard smartphone screen dimension.
So will dual screens be the "next big thing" or are they just another curiosity, a side trip on the technology path that will turn out to be a dead end?"
The marketplace will decide, of course. That's how it works.
Until then, dual screens are likely to be seen as a solution in search of a problem -- unless of course Apple suddenly decides such a phone would be a good idea.
But don't hold your breath.