Just brainstorming ideas:
Indiana Jones is older now. He's out of step with the 1950s, the era of big organizations--the man in the gray flannel suit, agency panic, the rise of L. Ron Hubbard's Scientology, post-war boom, and trauma. Indy isn't interested in keeping up with the Joneses or in going to get psychoanalyzed.
And what does Indiana Jones want? Jones doesn't usually want to be part of the adventure, he's either recruited by the government, falls into another adventure, or has to go rescuing his dad. For all his derring-do, Jones is a classic refusenik, as if all he wants in the world is to quietly collect a few artifacts and go on teaching, though he's a little old for the first and I don't see a lot of people painting their eyelids with "I Love You" anymore.
Well, what if we reversed that--what if Indy, afraid of being put out to pasture, was actively trying to get into the game? What if his safety net--Marcus Brody, the college, his dad--were all gone? Could he jaunt off and put himself in danger on purpose? Could he have a secret death wish--better to burn out than fade away? Might he already be dying of something or other? Cancer, heart problems? (After all, his mom died, probably of cancer. Note: Just did some research and she either died of scarlet fever or flu, so nix that.)
What would Russians do with some ancient alien artifact? Or is perhaps the artifact the guide to some secret cache of technology? Or perhaps the Russians searching for the artifact aren't loyal Soviets? The maverick Irina might be in it for herself, she might have some family legacy of state purges that she wants to get away from. Perhaps the maverick Indiana might find a partner in the maverick Irina, only part of his journey will be towards some form of community, while her interest in independence is unbending.
Theme: community and/vs. the individual? Indiana Jones, bereft of his community, has to rediscover the connectedness of people. Maybe the prologue should feature some failure--he fails to get the diamond, fails to save the cross from the rich speculator. Or does he succeed, but his associate lays down the "this is the last time, Indy" line. Indy may be experienced, but he's also kind of a jerk. With all the people who would put up with his issues gone/dead, Indy needs to figure out how to appreciate people.
People or just a few persons? A romantic angle? It makes sense for them to bring back Karen Allen, as the idea of senior citizen Indy picking up someone completely new might seem a little odd. But who says we really need a romantic angle: there was Karen Allen in the first, but his deepest relationship in the second is with Short Round, and in the third, it's his father.
A fight in a department store--there's a dig going on. A booby-trapped house: an Oxley-like ex-professor, exposed to the paranormal, started a cult, became too extreme. Wernher von Braun--"I was just trying to kill you a few years ago, now we're working together." Jet Propulsion Laboratories and the esoteric interests of Jack Parsons. Atomics. Space race. Lost Atlantis--aliens have been here before.
Mirrors of Indy: the Russian maverick who is more ruthless than Indy about preserving her independence; the superannuated crazy professor whose isolation has driven him to extremes (and vice versa); the hotshot expert kid who is too hotheaded to listen to others.
Problem: although Temple of Doom isn't as jet-setting as 1 and 3, there's some sense of jet-setting, and right now, most of my ideas revolve around America. (After all, most UFO sightings tend to be in America, a country sort of haunted by the idea of immigration and the future.) But if I want a more global setting, what? Easter Island? Nazca lines? Bermuda triangle? Tunguska?