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Not being a comics expert, I can't be sure, but I think this applies mostly to Marvel comics.

Why do so many characters have double initials? There's Pepper Potts, Reed Richards, Sue Storm, Peter Parker and I think at some times there was Bruce Banner (when it wasn't David Banner).

Is this an inside joke or is there some reasoning for this?

Given the fact that Stan Lee was the writer and co-creator across so many of the titles and characters at the formative years of Marvel Comics, it was a way for him to make it easier to remember them when writing and fleshing out the scripts.

Here's a snippet of a Q&A with Stan Lee around the premier of Spider-Man 2 that explains it all:

And we can chalk up all the alliterative names in the Marvel universe to Stan Lee's one failing. "It would be hard for you to believe this, because I seem so perfect: I have the worst memory in the world," Stan said. "So I finally figured out, if I could give somebody a name, where the last name and the first name begin with the same letter, like Peter Parker, Bruce Banner, Matt Murdock, then if I could remember one name, it gave me a clue what the other one was, I knew it would begin with the same letter."

If you want to look at DC Comics, and in particular Superman, it may have been due to a personal romantic connection of Joe Shuster, one of Superman's co-creators:

What is the L. L. connection for Superman?

A remarkably large number of characters in the comics have the initials L. L. Most notable are Lex Luthor, Lois Lane, Lana Lang, and Lori Lemaris. A rumor says that these names were chosen because Joe Shuster's first girlfriend had the initials L. L.

WARNING: TVTROPES ALERT!!!

From "AlliterativeName" article on TVTropes (having a name with the same first and last letter is called "Alliterative Name"):

In comic books, this is especially true of the names of superheroes or their close hangers-on. It was a favorite tool of Stan Lee's, since, swarmed with projects, he often had trouble remembering the characters' names, and the alliteration worked as a mnemonic device. (Though it didn't always work perfectly — Lee occasionally referred to "Peter Palmer" and "Bob Banner".)

In comic books, the Alliterative Name is often also "Two First Names". In cartoons, it goes hand-in-hand with "Species Surname".

I think he just did that to remember easier and its cool Bruce banner, sue storm, reed Richards , peter parker , pepper Potts, and the one no body thought of THE THING

Although it is not unique to Marvel Comics, which means that Stan Lee's own poor memory is somewhat irrelevant, I would imagine that it did begin as a mnemonic device to facilitate easy recollection of the characters' names, which became a longstanding tradition in the comic book industry. In my opinion, it's just too common to be explained any other way:

Source

Believe it or not, this is nowhere near a complete list - for example, Vicki Vale and Beast Boy are missing, as are many, many more in the DC and Marvel universes. Other comic book publishers are no exception to the rule - e.g., Archie Andrews and Jughead Jones. Cartoon characters with alliterative names include Mickey and Minnie Mouse and Donald Duck, Fred Flintstone, Bugs Bunny, Daffy Duck, and Mr. Magoo. And long before any of these, we had Peter Pan.