The friction contrafibulator was a button on the console of the Doctor's TARDIS. It existed on both the War Doctor's console (TV: The Day of the Doctor) and the Eleventh Doctor's first console. (TV: Vincent and the Doctor)
Later, the Eleventh Doctor used the button to stabilise the TARDIS interior when the TARDIS was confused by the presence of the Eleventh, Tenth, and War Doctors. The Eleventh Doctor pressed the contrafibulator on the War Doctor's theme and the interior stabilised into the form of Eleventh Doctor's second control room. (TV: The Day of the Doctor)
Behind the scenes
Though the word is superficially technobabble, it's also a subtle connection to another Richard Curtis script: Blackadder the Third's Ink and Incapability. In that story Blackadder frustrates lexicographer Samuel Johnson by throwing out a few nonsense words — including contrafibularity — after Johnson has claimed that his dictionary contains "every single word" in the English language.
What is it?
Breaking down this non-word to its real components, we find that it means contra (against) + fibula (the smaller of the two major bones of the leg): "pulling one's leg".
One more meaning, more appropiate for the TARDIS, can be if the word "Fibulator" means a device which operates as a clasp (synonym of Fibula), something that fasts, links or locks one, two or more parts.
"Contra-", with the meaning of "against", is added to the word: so a contrafibulator operates in an opposite way of a fibulator. Moreover the whole operation is aimed at a friction.
Thus a friction contrafibulator is a device which removes or opposes a friction, maybe not the mechanical one. However this meaning has not been given by the Doctor and it's just a hypothesis.
Another way to break down the phrase contrafibulator is from "contra" (against) and "fibrillation" (irregular beating or contractions, as in a heartbeat). It's like a medical "defibrillator" that shocks the heart to make it beat regularly. So, a "friction contrafibrillator" would counteract irregular pattern effects due to friction of some sort.