Why is Mew numbered 151? Mewtwo is the clone of Mew. This means that Mew existed before Mewtwo. 

So why is Mew numbed 151 in the Pokedex and Mewtwo 150?
Let us be reminded that the Pokemon franchise started off from video game, and its initial release was Pokemon Red and Green in Japan in 1996. So Mew's very first appearance can only be in Japanese Pokemon Red and Green.
According to Bulbapedia, in the Trivia section of Mew:


Mew was a little-known secret when Pokémon Red and Green were first released in Japan. Even Nintendo was not initially aware that Shigeki Morimoto had programmed it into the game.

In an interview with Game Freak and Creatures, Inc. in Nintendo Power issue 134, Shigeki Morimoto revealed that he created Mew two weeks before game development was finished.



Quoting from the interview which is cited in the Bulbapedia page:

Morimoto: We put Mew in right at the very end. The cartridge was really full and there wasn’t room for much more on there. Then the debug features which weren't going to be included in the final version of the game were removed, creating a miniscule 300 bytes of free space. So we thought that we could slot Mew in there. What we did would be unthinkable nowadays!
Ishihara: This is in spite of being told after debug ended that you weren’t to tamper with even a single bit! (laughs wryly)
Iwata: What’s the point of going through all the trouble of the debug process if you’re going to go and fiddle with the game afterwards…? I’d venture that this all came from Morimoto-san’s mischievous nature.
Morimoto: Well, it was a prank that everyone right up to Tajiri-san was in on. But even though Mew was in there…
Iwata: …It wasn't actually supposed to appear in the game, right?
Morimoto: Right. Unless we could think about any good opportunity to do so, the existence of Mew wouldn’t have been revealed to the public. It was left in there in case it was suitable for some post-launch activity. But if there wasn’t anyone among ourselves who wanted to use it, I thought it would be fine to just leave it as it was.
Iwata: Mew could very well have ended up never appearing in the game.
Morimoto: Right. But then due to an unforeseen bug, Mew ended up appearing in some players’ games. It looked like we planned all of this, but that wasn’t the case. So although it caused all sorts of problems to many concerned, fortunately enough it ended up having a positive effect.
[...]
Iwata: That’s how the “Legendary Pokémon Offer”8 came about.
8 The “Legendary Pokémon Offer” was announced in the April 1996 edition of CoroCoro Comic. Twenty winners would be selected who could send their game cartridge in so that the Mew data would be uploaded onto it. There were approximately 78,000 entrants.

From the quotes above, it is evident that Mew was not supposed to appear in the first generation. It was only programmed in at the last minute, occupying the space of debug feature.
The interview hinted at the fact that Mew was added after the debug process had ended, so swapping the number of Mewtwo and Mew at that point would have been disastrous, since Mewtwo had an event in the Cerulean Cave, and its stats, move set, flavor text, ... record would have to be swapped with Mew's. Appending a record at the end was safer, especially when the record was not supposed to be used at all during the normal game play.