Note that as indicated in the original post these are not necessarily obscure, just movies that other people seem not to have heard of for various reasons.
Let It Ride - Comedy with Richard Dreyfuss, Buster Poindexter, Jennifer Tilly, Teri Garr. "I'm not gambling, I'm WINNING!"
Moon Over Parador - Dreyfuss again, and Raul Julia. Dreyfuss is an actor kidnapped and forced to play the part of a dead bannanaland dictator by the head of the secret police.
Always - A Spielberg movie that a lot of people seem to have missed entirely.
Secret of My Success - Michael J. Fox in his second best movie ever (Back to the Future being #1)
Fandango - an unknown early Kevin Costner flick costarring Judd Nelson. Vietnam-era college buddies on a post-graduate road trip. It's a full-length movie built around a student film by director Kevin Reynolds.
The Beast - another Kevin Reynolds directed film. A Soviet tank in Afghanistan pursued by Mujahadin.
Postcards From The Edge - behind the scenes Hollywood stuff based on Carrie Fishers book.
Once Were Warriors - industrial strength depression from New Zealand. Stars Temuera Morrison, better known these days as Jango Fett.
The Grifters - neo noir with John Cusack, Angelica Huston, and Annette Bening
The Player - More behind the scenes hollywood in a dark comedy/suspense vein.
The Man Who Would Be King - Easily overlooked Sean Connery and Michael Caine film based on Rudyard Kiplings story.
Breaker Morant - Australians fighting in the Boer War accused of murder. Edward Woodward, Bryan Brown, Jack Thompson.
Hope and Glory - John Boormans autobiographical account of the home front in England in WWII.
Empire of the Sun - Another often overlooked Spielberg movie about a British kid seperated from his parents when the Japanese capture Shanghai in WWII.
Mystery Train - Jim Jarmusch directed. Several stories all set in the same fleabag Memphis hotel on the same night.
I'll also second nominations for:
Comfort and Joy
Local Hero
Last Days of Disco
Rustlers Rhapsody
Night on Earth
Into the Night
Raising Arizona
Repo Man
-any movie with Gene Hackman.
Edit: added a bit more. I could go on but I think that's enough for now.