I was never big on the 16-0 banner, (not embarassing, just not a banner accomplishment), but I do think teams should hang Conference Championship banners. In the US, we all love the all-or-nothing nature of the sport, but in many sports leagues there are actually trophies for best regular season record, cups for tournament wins (soccer.). I don't think that banners should be overdone (the Colts, the most glaring example), but I do think that a conference title is something to be proud of. Divisional championships are probably too weak with a mere four team race, although listing the number of divisional titles on one banner wouldn't be that ridiculous.
Anyway, back to the Conference titles. You get the Halas or Clark trophy for winning the conference. The AFC and NFC both have storied histories, rivalries, etc. The conferences are clearly divided into two distinct fields of competitors. And making a Super Bowl appearance is an exceptional accomplishment. Being in the Super Bowl as a participant, win or lose, makes you part of history.
So I think the Pats should have the following banners:
*1 white banner that says "19X Divisional Champions"
*9 red banners that say "AFC Conference Champion" for 1985, 1996, 2001, 2003, 2004, 2007 (could add a small marking for undefeated), 2011, 2014, 2016
*5 blue banners (which are much bigger and more glamorous like the current ones, showcased from the rest like they are now) that have SBs 36, 38, 39, 49, 51
Just to emphasize, I wouldn't want the stadium to be overwhelmed with non-championship banners and would not be the center of attention. The conference championship banners would not be as big as the SB ones. I think the Pats should be proud to have more divisional championships than any team in the AFC East and more Conference Championships than any team in the NFL.