"Our mutual friend" is a code phrase used by the main characters in Person of Interest to avoid referring to either Finch or Reese by name in sensitive situations. It has also been used by Root to refer to the Machine. It is a commonplace American expression, but may also refer the English novel of the same name.
"Our Mutual Friend" is the last novel completed by Charles Dickens. It is widely considered one of his most sophisticated and complex works, combining psychological insight with social analysis.
Unlike other works of literature, the book has never appeared on screen, but was referenced by a drugged Finch during a phone conversation with Reese. Finch says, "The plot turns on a case of concealed identity. How ironic." (“Identity Crisis”)
Finch: Our mutual friend went after your perpetrator, and I haven't heard from him. Something's wrong.
“Wolf and Cub”
Fusco: All right, look... First you got me investigating our mutual friend. Now I got to babysit this disrespectful punk?
Reese: Finch, I've got this end under control. We're preparing a surprise for our mutual friend.
Finch: Our Mutual Friend, Charles Dickens, published... 1864, '65. The plot turns on a case of concealed identity. How ironic.
“Many Happy Returns”
Finch: Well, if there was any evidence that threatened our mutual friend, I'm confident that you would find it before Donnelly would.
Carter: Are you suggesting I obstruct a federal investigation?
Carter: Let me guess, I have to cancel my dinner plans.
Finch: Our mutual friend needs help, detective.
Fusco: Have you heard from our mutual friends?
Fusco: Glasses sent me out looking for you and wonderboy last night.
Carter: Oh. Our mutual friend.
Fusco: Looks like your day job just began.