Question: I'm currently working on a short story in which the main character has received a device from an anonymous individual. This device *appears* to function like a regular phone. Until he realizes that he can send text messages up to 1 hour back into the past. 

My question being... How would such a device (theoretically) function?

Note: Handwaving is allowed but do NOT make your entire answer one big handwave

Answer: 
Wormholes.
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Wormholes are hypothetical "connected black holes". In essence you have two points in spacetime which are always connected to each other, and matter and information could *potentially* pass from one to the other very quickly (apparently faster than lightspeed) without violating relativity.

In real life we don't know if these actually exist, and even if we had one we don't know any way to put something through without it being obliterated. (After all, you just threw something into *a black hole*.) But since you're writing fiction you can ignore both of those points. More importantly, it hasn't been proven impossible; it's conceivable that in the future we'll discover stable wormholes and find a way to send something through.

So suppose the creator of this telephone (a mad theoretical physicist on the Space Station who's a big fan of *Steins;Gate*) discovers a tiny, stable, traversable wormhole. He doesn't want to publish this absolutely phenomenal discovery until he's sure he really has found a wormhole. So, being mad, he decides to try a dramatic demonstration.

The wormhole mouths have happened to move relative to each other in very contrived and convenient ways in the past. Now they're just over an hour apart in time. (Relativity can do this; see the "Twin Paradox".) So our mad physicist builds small "relays" which orbits the mouths, somehow not being destroyed by the gravitational forces.

This special phone is capable of sending very short messages to the relays. When it does so, the first relay sends a signal through the wormhole in the present. The signal leaves the other mouth one hour in the past, from the sender's perspective; the second relay records it and transmits it back to the phone on Earth. And when the phone receives a message back from the relay, it sends it as a standard text message to the specified number.

The physicist tests this...and it works! He can send causality-violating messages (within the confines of [Novikov's Principle](https://en.wikipedia.org/wiki/Novikov_self-consistency_principle))! This is certain to win him that Nobel he deserved twenty years earlier! But then, being as absent-minded as he is mad, he accidentally loses the phone.

To anyone else picking it up, the phone seems magical. Send a text message and it arrives one hour in the past. Simple as that. This is the best method I can think of for not *blatantly* violating current physics.
It's not possible
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Imagine this:

A phone (let's call it phone A) gets destroyed. 30 minutes later, the *device* sends a message to phone A 1 hour in the past. There is *no* way that phone A could receive that message since it already got destroyed. It can no longer receive any message ever since it got destroyed, even if that message was sent before it got destroyed. This is the same problem with time travel. You cannot bring a dead person back to life in the real world.