Why did Cessna use a new design for the C162, instead of updating the C150 design? As I stare at the ceiling unable to fall asleep, airplanes are on my mind.  Failed airplanes.  Specifically: the Cessna 162.

I understand that Cessna perceived a need for a two seat trainer.  The CEO himself called that out: "The need for a modern, cost-effective two-seat trainer aircraft has never been greater, and we believe we are well positioned to meet that need."  I also understand that the light-sport aircraft (LSA) market was thought to be a significant driver for aircraft sales.  That turned out to be a gross miscalculation.  Cessna ended up selling 192 Skycatchers (versus a planned run of 600 per year) due to quality and price misses.  About 80 were not sold and retained as parts donors.  A fleet of ready-made LSA aircraft already existed (Champs, Chiefs, Cubs, T-Crafts, etc) so the value proposition of a new $150,000 LSA runabout was not compelling...or so it seems.

The following question assumes that there are folks on SE.av that have some industry knowledge of the GA manufacturing scene...

Why did Cessna not just resuscitate the proven 150 line (~24,000 built) and forego the LSA segment, given that the objective was to capture the two-seat training market and a good number of very cheap LSA airplanes already existed?

I recognize that there may be no good answer for this.  Maybe it was just one of those plausible-at-the-time business moves that didn't pan out.
I think there are a couple reasons: 1) Weight, and 2) the Type Certificate.

For #1, the original C-150 is roughly 100 lbs too heavy (in empty weight).  That's a bit over a 10% reduction, which is huge in an all-aluminum design.  Note that later models need even more weight removed (C-152 included).

For #2, the C-150 is a CAR Part 3 type certificated design.  To sell it as a LSA would almost certainly require a re-designation so as not to conflict with the original type certificate.  This is primarily a paperwork thing, but nobody had done it before (even now, none of the "grandfathered-in" type-certificated designs are still built under their type certificate or by the original manufacturer).  The C-152 is an add-on to the C-150's type certificate, so the same problem applies to it.

As far as why the C-162 failed, I think Carlo's answer covers most of the reasons pretty well (ignoring the "GA is dead" rant), though I'd like to add that Cessna did a terrible job with designing the C-162 and took so long to "get it right" that they didn't start deliveries (of an arguably inferior product) until right after the 2008 economic downturn.