I suppose the most recent rare car I've seen made its first public appearance last year: the Porsche LMP2000.
After winning the 1998 Le Mans 24hrs with the GT1-98, Porsche took a sabbatical to begin work on a successor with the aim to return with full force in 2000. The initial powertrain was a derivative of the 911's turbocharged flat-six which was ultimately deemed inadequate for the task, so instead they redeveloped their unused 3.5L n/a V10 that was originally intended for use by the Footwork Arrows F1 team in 1992, as a replacement for the 3.5L V12 unit from the previous year. The capacity of the V10 was increased to 5.5L, a variety of F1-specific components were removed, and a new chassis to house it in was built from scratch. The car never actually received a name: LMP2000 is the widely accepted name but it was only ever known internally by its code name 9R3 (9R6 was the RS Spyder, and 9R9 was the 919 Hybrid).
By the middle of 1999, the project came to a halt and was eventually cancelled, but the car (the only one in existence) was allowed to undergo a two-day shakedown test at Porsche's Weissach facility with Bob Wollek and Allan McNish sharing driving duties. The official reason for the cancellation of the project, confirmed in December 1999, varied, with the most widely noted being to divert resources toward development of the Cayenne. Today it's accepted, and confirmed by insiders, that a deal was struck between Ferdinand Piech, then chairman of the VW-Audi board, and Porsche's Wendelin Wiedeking to co-operate on the SUV project (which in turn led to the creation of the VW Touareg and the Audi Q7: they both shared a common platform with the Cayenne). At the same time it was agreed that Porsche would halt its LMP project, and withdraw from top level sportscar racing, thus paving the way for what would become 13 outright victories for Audi at Le Mans from 1999-2016. 9R3 eventually found its way into storage in Stuttgart, with knowledge of its existence denied by many high ranking Porsche officials for a time. The V10 engine however would have a different end to its story.
As a way of self congratulation for their hard work, the engine was put into a concept car and put on display at the 2000 Paris Motor Show. Interest in this "new" car far exceeded Porsche's expectations, and with revenues on the up since the launch of the Cayenne in 2002, the green light was given in 2003 to put the concept into small number production. With the engine modified once more to 5.7L in capacity and producing 605bhp, the end result, launched in 2004, was the Carrera GT.
As for 9R3, its existence was finally acknowledged by Porsche a couple of years ago, and as part of its 70th anniversary celebrations in 2018, brought the car out of storage to be shown to the world for the first time at the Goodwood Festival of Speed.