Beckett successfully time travels, but finds that he is living the life of someone else in the past. Able to communicate through time via brainwaves, Beckett and his holographic companion Al determine that Beckett has been intercepted by an "unknown force to change history for the better."
In the reboot, these things are still in place, but Sam makes more of an effort to gain control over the leaps and in doing so, he is continually put in his place by the "unknown force." Initially wanting to make it back to his own time, Beckett accepts his mission to positively influence history with the hopes of meeting the entity sending him on these missions. The series would be episodic for the most part as was it's successor, but would have an over-arcing story showing Sam struggle with accepting his fate.
The original series introduced the concept of "evil leapers" and the reboot would eventually get around to that culminating in the big reveal that Sam's quantum leap experiment actually killed him and there is no returning home ever. He is, in essence, stuck between life and death being used as an agent of change for a good force combating agents of change operating on behalf of an evil force.
The original series focused on the 50s, 60s, and 70s with occasional visits to the 80s. Since Scott Bakula was born in the mid-50s, it made sense that his Beckett's lifetime would cover that era. For this show, I propose casting someone younger and setting the show in "the near future." Jon Hamm, born in 1971, would take the lead role giving the rebooted Dr. Beckett a lifetime that begins in the mid-70s. The time period would focus on the Gen X years rather than the Baby Boomer years (70s, 80s, 90s).
Again, Beckett is joined by an observer from his own time who is initially known only as "Al." In the original series, Al was an admiral in the Navy who somehow ends up working as a lackey for a civilian scientist. That doesn't make sense. In fact, I'm not sure how the Navy really has anything to do with this project so I'd drop that whole angle. I'd also want to do more with the "present day" stories and expand the operations of Project: Quantum Leap. In Dr. Beckett's absence, the government has sent someone in to take over. Al must continually keep the project from getting shut down. My castling choice for Al would be Benicio del Toro, although, that might be a tough guy to get.
The original show was rather pricey, not just because of special effects, but because of it being a period piece every week. I'd not want to lose any production value and would rather this reboot be done in shorter seasons with greater focus given to each episode.