Eye of the storm

Eye of the storm was a phrase referring to people not affected by an event, such as a change in time, due to them being at the very centre of the catalyst of that event. (TV: Last of the Time Lords, The Big Bang, The Lie of the Land)

History
During the Last Great Time War, the Daleks were able to break through the quantum shield protecting Earth in 1961. Emerging in East Berlin, they proceeded to exterminate humanity. However, the Daleks were thwarted by the War Doctor, who shifted the events of the invasion into an alternate timeline. Nevertheless, he acknowledged that East Berlin was the "eye of the storm", and so certain individuals would remember the encounter with himself and the Daleks. (AUDIO: The Shadow Vortex)

As noted by the Tenth Doctor, the aircraft carrier Valiant, having housed the Paradox machine created by, was the "eye of the storm" of the Year That Never Was. As a result, everyone who was on the Valiant at the time of the Paradox Machine's destruction remained and remembered the events of the past year despite the reversal of time. (TV: Last of the Time Lords)

When the Doctor's TARDIS was destroyed by the Silence on Earth, the explosion caused a Total event collapse which erased the vast majority of the entire universe from history. However, Earth would remain as the "eye of the storm", with the Eleventh Doctor explaining to Rory Williams that "[they were] just the last light to go out." (TV: The Big Bang)

As the Twelfth Doctor explained to Bill Potts, she was unaffected by the Monks' influence on human minds while within "Fake News Central" because they were at the "eye of the storm", since all the lies were broadcast from there. (TV: The Lie of the Land)