1. Warp Drive
The Warp Drive is the big deal, and if we ever do manage to achieve this as a species then we can leverage a fair deal of credit at Star Trek's door. Sure faster than light travel has been covered in science fiction since the genre began, but Star Trek didn't aim for the simple "light speed" travel (although in the original pitch, this was what Gene Roddenberry went for). At some point during The Original Series, warp speeds (and the various degrees of it) were invented. In the NASA science that has used Star Trek as an influence, the theory is that you warp space itself. Thankfully this is simply called a space warp rather than something incredibly techie. It would have the effect of shrinking the space in front of the spaceship and expanding the space behind it, allowing the ship to move faster than the speed of light without actually doing it because they've moved a much shorter distance than it actually was. Propulsion researcher Harold White compared this effect as to stepping onto a moving walkway at an airport but maintaining the same walking speed. Technically this means that the person is still moving at a walking speed by their perspective, but to everyone else's perspective they would be moving faster. Of course managing to develop requires several other major breakthroughs such as gravity control, and of course some sort of ability so that you don't just fly straight into an asteroid. Of course these technologies are already in place in Star Trek allowing it to do this. By using this ability you don't so much as break the speed of light as you would break the sound barrier, but instead side step the issue. Of course the actual speed of light is theorised to be relative to the observer's frame of reference. Don't just ask me that's what Albert Einstein says. So warp speed may end up being set up much like in Star Trek with number assigned which mean a pre-determined speed itself rather than claiming any specific speed is equivalent to the speed of light. Mind you, not a scientist or expert in this area, so anything I say about NASA research may already be out of date! :)