A great Indy game needs to be first and foremost a solid story-based game with strong characters and a good sense of humor and intrigue. It should unfold like a movie--Uncharted is a good example of a game that did this well. Beyond that, it absolutely needs to be 1 part Tomb Raider Legend (with whip and revolver instead of grappling hook and silly dual pistols), 1 part brawler, and 1 part inventory-based adventure game (like Fate of Atlantis.)
What it does NOT need to be is a broken jumble of different minigames strung together with a poorly fleshed out plot, like Staff of Kings, or a game that is ALL brawler with no substance or puzzles like Emperor's Tomb. I don't necessarily think it needs the epic, sprawling locations of a game like Assassin's Creed (since when has an Indy movie really had these kinds of locations anyway?) I also don't think the mind-numbingly difficult platforming action of the SNES game (as good as it was for its day) is something a new Indy game should seek to emulate.
Infernal Machine was pretty good story and exploration-wise, but lacked the action of the movies (or a control scheme that was suited to action at all)--and the story kind of fell apart towards the very end, as well.
Fate of Atlantis had the kind of fantastic movie-quality story and script that would be needed for a great Indy game, not to mention great inventory puzzles, but due to the limitations of its day, didn't really have much of the action elements that a game would need to compete in today's market.
I would also love an Indy game that truly went for the whole "action set-piece" aspect of some of the scenes from the films. I mean, how great would it be if you could tackle something like the Raiders truck chase in game, having the option to solve the problem in a number of different ways, one of which could involve getting thrown out the windshield and using your whip to hang on to the bottom of the truck for dear life. Or figure out how to create a distraction and get aboard the Tank to rescue Henry Sr, but then actually have to do all of the action to get there, like riding the horse, clogging the guns, jumping aboard, fighting Vogel, etc. That kind of stuff needs to be translated into video game form. Video games as a medium are at that level where they can really tackle those kind of scenarios in game, and do them well--the idiots over at Lucasarts just need to figure out how to do it with Indy.