The Bunbunmaru Newspaper (文々。新聞) is run by Aya Shameimaru.
文 (Readings: "Bun", "Aya", "Fumi", "Mon", or "Kazaru") is a Japanese kanji that is generally used with regards to literature or writing. For example, "文" is used in compounds that mean letter, character, sentence, word, literature, writing, culture, etc.
As such, it is difficult to precisely translate the meaning of the newspaper's name to English. Perhaps an effective way to understand in this case is to combine all the meanings previously listed and think of it as an ideogram that represents those ideas.
The standard reading of "文々。新聞" is "bunbunmaru shinbun", meaning "sentence sentence period newspaper".
One simple explanation is that because Aya's name written in kanji is also "文", the newspaper is named after herself.
Although not canon, given the "Aya" reading of "文" (the same kanji used in Aya Shameimaru's name), the paper's name could also be read as "Ayaaya Maru" in possible reference to her catch-phrase from Mountain of Faith. The phrase "文々。新聞" when read as "Aya Ayamaru Shinbun" is further a homophone for "文誤る新聞" (meaning "Aya Misleads Newspaper" or "Aya makes Mistakes Newpaper") or "文謝る新聞" (meaning "Aya is Sorry Newspaper" or "Aya's Apologetic Newspaper"). But none of these readings are conventional or official.
Another interpretation is that Bunbunmaru could be one of ZUN's jokes. There was a Japanese baseball player, Takahiro Ikeyama (池山隆寛). He was a player with particularly memorable characteristics. He got many homeruns when in good condition, but he sometimes got many strikeouts in bad condition. So he was given a nickname Bunbunmaru (ブンブン丸). "Bun bun" sounds like two bat swings, and "maru" is an ordinary, albeit old-fashioned name ending for a man. There is one more man having the nickname, but Ikeyama is far more popular. It is possible that ZUN created this joke to be associated with Ikeyama.
Many issues of the Bunbunmaru are included in Bohemian Archive in Japanese Red.