Quantum Leap/WMG

Sam Beckett is going to replace everyone on Earth.
Since he never returned home, he is going to jump for eternity. Therefore, he is going to replace everyone in the project's timeframe. There is a flaw in this reasoning, though. The only person he isn't going to replace is actually himself.
 * Wasn't there an episode where Sam leapt into his teenaged self and attempted to prevent his sister's abusive marriage and his brother's death in Vietnam?
 * He'll never leap into himself in his own time.

Sam Beckett is altering his own abilities.
When Sam changes history, he also slightly alters his own abilities. That explains his unusual knowledge. That and he is a genius.

Sam is what happens when a human time lord dies.
Nobody has ever seen what happens when a human time lord dies. In Family of Blood, the Doctor only has a vision of his future self, and it doesn't show him dying, and when The Master becomes human, he opens the watch before dying. In the original series, when The Master ran out of regenerations, he flew around as a pool of blood, possessing people. Sam was a Time Lord that was humanized and sent to earth, with Ziggy a front for his TARDIS (but not the TARDIS itself) and would have regained his time lord attributes had he not killed himself a dozen times over trying to activate Quantum Leap. To prevent him being found out, the Ziggy sends him back to keep him alive until Gallifrey can be contacted, and helps people in the meantime (see the TARDIS Is The Hero guess on the main Doctor Who WMG page). The TARDIS reconstructs the personality of the character and runs them through a simulacrum of Doctor Beckett's body, and the real person's mind is there the whole time, influencing Sam's behavior and remembering the things they did while under his influence. Al isn't actually linked to Sam, it's just that the TARDIS doesn't trust anyone else enough with communicating directly with Sam's mind.

The G-Man is leaping Sam around.

 * DOCtor BECKett is...not AS hsssh...obsERvant as MisTER Freeman and SIMPly haSn't...noticccced.
 * Is that the G-Man, or Torgo?
 * Perhaps Torgo the White.

Our current time line is the result of Sam's actions.
He comes from an advanced future compared to us around that time frame. We are slowed technologically by the effect of many, many, many minor changes Sam does to make the future "better". Yes, he improves individual lives, but is slowing technological progress in the process. Imagine how many of those lives he "saved" could have spurred those around them to do greater things had the tragedy happened?
 * This is actually backed up in the series itself (at least, the "Sam's creating our timeline" part - the technology difference between timelines seems to be confined to the technology involved in Project Quantum Leap itself, and we all know Reed Richards Is Useless). In the episode where he thinks his mission is to save JFK, he fails, but then learns that his mission was actually to save Jackie Kennedy who, according to his and Al's history, originally got shot and killed too. Since she survived in our timeline...

Sam's Meaningful Name tells us he never was going to get home.
He's named Samuel Beckett. He's no more likely to go home than Godot is to meet Vladimir and Estragon.

Sam was once abducted by the Great Race of Yith.
As with all their research projects, they used their alien technology to leap one of their researchers into Sam Beckett's life so they could investigate 20th century humans, while he found himself living in their era as a captive but honored guest. After months of learning about the Great Race and its leaping technology, the mission ended and Sam leaped back to his own body again. But their Laser Guided Amnesia developed a few cracks, and Sam gradually, unconsciously remembered enough about leaping to invent his own version of it. Alia the "evil leaper" and her dystopian world are from an alternate future ruled by a rogue, human-leaped faction of the Great Race that worships Azathoth, and is trying to destabilize the timeline and bring about chaos.

The reason Sam never makes it home...
...is because he ends up inadvertently erasing himself from history or paradoxes himself out of existence (or, accidentally kills his younger self, like he did with Al in that one episode). This also means all the other leaps get "reset" and now everything happens as it did the first time around.
 * Alternatively, we don't really know how Time Travel works in the Quantum Leap 'verse; it seems to be For Want of a Nail since the Quantum Leap Project still exists no matter how the past has changed, but the Future Computer still somehow knows what the "original" history was before Sam changed it. For all we know, Sam's been dimension-hopping and he's somehow accidentally erased the way back home--or, he rejected the possibility of going home so he's made a significant difference in at least one timeline.

Sam is God.
This is a simplified version of an IRC discussion that involved This Troper and several fellow online Leap fans.

Okay, so Shinji, Haruhi, and Pickle Inspector all got to this one first. Think about it: the dominant theory in the first part of the show is that "god" controls Sam's leaps, and we later learn that Sam is doing it subconsciously. But if Sam never returns home, he leaps infinitely and thus has complete control over the timeline after his own birth. This gives him control over the circumstances of future Time Travel projects, which means he can theoretically leap into other time travelers, allowing him to go further into the past and influence the origins of the universe. Thus, Sam has complete control over the universe.

Wishbone is Sam's dog.
When Sam Beckett got stuck jumping through time, his dog was in the machine, too. Only the dog, Wishbone, ended up jumping through literary characters instead of actual people. Unlike Sam, Wishbone did eventually leap home and was adopted by some kids, to whom he attempts to impart the wisdom of his travels.

Al retains both his original and altered memories.
The end of his marriage to Beth was a character-defining moment in the original line, informing everything his did afterward. Since there's no indication that he doesn't still join the project in the new timeline, he and Sam must still be friends. The fact that he remembers the original line might contribute to the success of his marriage in the new one. If you remembered a life without your loved one, wouldn't you appreciate having them all the more given the chance?

Sam did get home.
Lothos hijacked the broadcast. Once Sam accrues sufficient Leaps to push the timeline away from where Lothos wants it, they'll face off. One shall stand, one shall fall.
 * This is now my personal headcanon.

The events of "The Boogieman" weren't all just in Sam's head.
Besides, of course, the bookends of him before and after he fell down the stairs. What happened was that Sam was there to save Tully, but the Devil was preventing Al from getting a lock, so Sam was unable to do what he was supposed to do. The Devil then prevented Sam from leaping around so that he could keep screwing with him. Eventually, once Sam vanquished the Devil, he immediately leaped back to the beginning right after he fell down the stairs and rewrote the episode's history, including Al's.

The reason time travel mechanics change arbitrarily...
Is that Sam's changes to the timeline affect the original project and retroactively make them different.

Ziggy is the one controlling the leaps.
And has a wicked sense of humor. Sam gave Ziggy a personality, so like another computer, it's playing a game with Sam, testing his mettle. That's why Ziggy's prone to 'make mistakes'- it's actually screwing with Sam and Al. Also, Al the bartender from the final episode is actually AI, a computer generated avatar for Ziggy.