Okay, we all know how Sony just let the Vita crash and burn.... But I do have these idea's that could save Sony's "handheld fanbase".   

They should design it specifically for Remote Play and Streaming games: Remote play has to be one of the most innovative features in our gaming generation. This feature helped the Vita sell a little better since it was introduced. This feature has the feature to connect over the internet to your PS4. THIS IS PERFECT for the next handheld device. They should have a 2nd control option for the PS4 games so the handheld users can play over the internet via wifi. I think the whole "PS Now" is gonna pick up just for 20 dollars a month. With the new handeld they can play PS1/2/3 games on the go with the PS Now. We need to make it the best experience as possible for both Remote Play and Streaming, so they need to FOCUSSLY design the handheld to play those features as best as possible.       

They don't have to create or develop games just for the new handheld: Developing games for a handheld is way too expensive and a waste of time, espically when the PS4(and 5) is selling like crazy. So they DON'T have to make games for it. They can create games if they want to, but why would they do that?

Have them run on android so the handheld can have access to the Play store to get those mobile apps!: Since the Mobile gaming is killing the Handheld market... How about the Handheld market join it? It's not rocket science, the PS Vita had a touchscreen as well, why didn't Sony do this earlier?? It's NOT too late! Let the next handheld system run on Android to have those gaming apps! I would sure play those apps on my Sony handheld device instead of my phone at home! Even if kids of these days don't have their own phones, they can play the trending app games on their handheld system!

Make the handheld device powerful just to play games and run smooth apps: They don't have to go all out with the most powerful handheld device at all. Just make it powerful enough so we can have a good experience! When we Stream or Remote play, or watch Netflix we would want that to run smooth as possible.

I can't believe I'm saying this... but put a good camera on it too!:  There's alot of good looking human beings out there who LOVE taking selfies... Make the handheld have good front and rear cameras! For Skype, ooVoo, and even broadcasting when playing a game.

Make the next handheld have a PS Vita game slot so we can have backwards compatibility: Everyone got pissed when the Vita wasn't backwards compatibility physically.

NO EXPENSIVE MEMORY CARDS!!!:  This is one of the damn reasons what killed the Vita!! My parents got me my Vita for my 13th birthday but I was still angry how expensive they were. PLEASE have built in storage!!  

Well there you have it... A device that doesn't need expensive games to make, made just for streaming and remote play to play games that are already made,  no expensive memory cards ect. Everyone wins with saving money..... What do you guys think???? IF YOU AGREE WITH ME, LETS MAKE THIS FAMOUS SO SONY COULD GET THESE IDEAS'S AND US ALONE CAN SAVE THE SONY HANDHELD FANBASE. WE WILL BE KNOWN AS THE HEROS IN THE GAMING INDUSTRY!!                                     

Edit: I notice I kinda put the wrong title for this.. but I got your attention. The new title is  Ideas that Sony should do to make another Successful handheld system.  

You know what would have sold some Vitas? Perhaps a GT and paid money to Rockstar for a GTA instead of going for a crappy CoD port.  Losing MH to Nintendo was the final nail in the coffin. 

None of these are reasons why Sony should get back into handheld gaming. These are suggestions for if they do. Some of which are terrible. "You don't have to make games for it". You basically described a phone

uknow the Vita sounded dandy until the price came into the equation. Not as attractive anymore eh?

Better camera? Make handheld powerful? Those cost money. Like, a lot of money.

I rather they make a 2in1 device, thats a smartphone AND a handheld in one.

physical buttons/dpad + L1/R1 ontop of it.
Bring out the Playstation name for the phone.
Make sure its cheap enough to produce, and uses normal storage.

Playstation Phone.

A vita phone would require a monthly subscription to work as a phone. If you're gonna go in that market, itd be a bad idea to have the form factor be a handheld controller with giant physical buttons and a thick housing

Sony already has the best mobile device on the market. The 3G Vita already does much of what you described, and Remote Play over 4G and Wifi has been possible since PS4 launched. The Vita already is the device just powerful enough to run apps and streaming services smoothly.

The problem is people refuse to see PlayStation as a device beyond gaming, and gamers attack any motion by Sony to expand the brand and grow the userbase. The gaming community is shooting itself in the foot with the bone head ways they have reacted to the Vita and Xbox One.

The Xperia Z line is pretty much the logical evolution of the PS portable line. I mean the Z4 onwards all have remote play built in. They're also the best phones on the market in almost every spec.

But guess what? Sony barely supports THAT either! They never put an aggressive marketing campaign and they've barely broken through with US carriers, so the line has a paltry marketshare.

So, to answer the OP, they pretty much already did the Vita successor, and they gave up just as quickly.

Let's leave them to the PS console line, shall we? They're a very fickle company as it is, and we don't want them to waste any more money than they already do on markets they're not going to actually try to perform in.

I just had this weird dejavu from reading your comment. 

so basically make a mobile phone that can stream games to it? yeah they already did that with their smartphones.