This is based off of an entry on the Doctor Who Fridge Logic/Brilliance/Horror page on TV Tropes. Look from the bottom up and you should find it pretty quick. The reference to the Storm Cage as a prison planet is based off of a snippet on the "Storm Cage" entry on the Doctor Who wikia; it says that in an interview Alex Kingston (the actress who plays River Song) thinks the Storm Cage may be a prison planet. I own nothing.
Many assume that the Storm Cage is River Song's prison. It's reasonable to make this assumption; after all, the Storm Cage is a prison planet and the only people who are ever on prison planets under normal circumstances are the staff, the guards and the prisoners themselves.
They're wrong.
Well, scratch that. They're not entirely wrong; they're about half right but that makes them half wrong as well. The Storm Cage isn't exactly River Song's prison.
It's her home.
River was never formally charged with the murder of the good man she killed. It's not like she ever denied her culpability in the crime, and she was too much in shock after the fact to resist arrest. River was never tried, never sentenced, despite what most believe. She's not even a convict, not officially, though she still waits for a pardon.
Her residing in the Storm Cage is as much for her own safety as for the universe's at large. River Song is without a doubt a danger to everyone around her but the point stands that River has Time Lord DNA along with her human DNA, albeit primitive Time Lord DNA. River's said it herself; empires would tear whole worlds apart for just one cell, and given her own existence the powers that be don't like to think about what scavengers could do with those cells. River's "imprisonment" is as much for her own safety as the universe's.
The guards don't like it when River leaves. How can they? Over the years she's mercilessly shredded the reputation the Storm Cage spent centuries cultivating; she practically runs the place! They don't like it, but they've never tried to chase her or track her down. They know better than to bother. River always comes back.
The cell River calls home is stacked with amenities no other prisoner can ever hope to see. Posters, books, candy hidden under the mattress along with the journal, hidden like a school girl sneaking snacks in class. The guards tried to take them away once. They learned their lesson. All those things she had brought, they were just back in her cell within the space of an hour. A very polite note was left ("Try that funny business again and die. This is Doctor River Song writing, thank you.") and none of them knew what to do, so they didn't try again.
When she's not escaping, River Song is a model "inmate". She's polite, friendly and genial. The guards all know why. They all know that she knows she's more than capable of leaving any time she wants to. Between the hallucinogenic lipstick and the Vortex Manipulator she acquired, the Storm Cage might as well be made of air. Sometimes, they wonder why they even bother to lock the door.
After all, River can get out any time she wants.
And she always comes back.
She was a child of war, bred for battle and nothing else. This is the only place River Song can ever possibly call home.
There's nowhere else for her to go.