Grateful Dead lyricist Robert Hunter wrote the words for this song, as he did for most Dead songs. Jerry Garcia wrote the music.
For years the legend was that "Bertha" was named after an old fan (an actual mechanical fan, not a fan of the music) that would jump around the Grateful Dead's office when turned on. David Dodd at the official Grateful Dead site clarified that old myth
by citing an interview in which Hunter revealed that the Dead named the rambunctious fan after the song, rather than vice versa.
What "Bertha" is actually about, Hunter believed (he claimed he couldn't remember for certain), is "some vaguer connotation of birth, death and reincarnation. Cycle of existences, some kind of nonsense like that."
That origin story could make "Bertha" a pun for "birth." It definitely opens up fascinating new possibilities for the song.