I think everyone knows what intuition means. For a psak, it translates into giving an answer without looking anything up or running all the relevant sources through one’s head before answering. I know of great poskim who acknowledged that they pasken this way. I won’t give their names.
Historical factors. I’ll give two examples. After the Chmielnitzki y”s massacres (1648-9), the rabbonim got together and devised ways to be mattir as many agunos as possible. If one reads of these massacres (Nasan Hanover in Yven Metzulah, Yaffa Eliach in There Once was a World, etc…) one will understand the importance of history in psak. Same for sociological factors in this case. One can only imagine the pain and agony of the agunah who pleads her case in front of the Rav, who is suffering himself no less. Do these factors affect psak halachah? Absolutely.
Second case. The Chasam Sofer is famous for increasing the level of issur for certain issues due to his battle with the Reform movement. One has to read the history books to fully understand this movement, and I mean from both sides of the coin. The sociological factors in the Chasam Sofer’s decisions are obvious. Issues like bris milah, halanas hames, all needed to be considered according to the social attitudes of the times.
I would say that almost all psak has sociological factors built in without them having to be spelled out. In our days, can anyone deny that Internet use and its’ permissibility or non-permissibility is at least partially decided by sociological factors? Or college and secular studies? Or b’nei chutz la’aretz making a public yom tov minyan in Israel when everyone else is putting on tefillin and going to work? That’s what I mean.