Martha is only mentioned by name three times—and near the end of the novel at that—and her identity is not explicitly explained, so readers have to use context clues to puzzle out her identity.
When Axel thinks of Martha, he is remembering what he calls the "beautiful house" on the Konigstrasse, where he lived with his uncle. Axel is at this point on the raft with his uncle and Hans, being whirled upwards toward the top of a volcano. Not surprisingly, he fears he will die. Therefore, he is fondly recollecting the life he believes he might be forever leaving behind.
He thinks of the "good Martha," without specifying who she is, but if we look back to the beginning of the novel, we learn that there were only four residents of his uncle's house: himself, his uncle, his uncle's goddaughter, and an "old cook." By process of elimination, we realize that Martha is the old cook, the sole servant in the household.
When he does get back home in one piece, Axel speaks of the "astonishment" of Martha to see him alive. He also mentions that due to her gossiping about where they have gone, the news of their journey to the center of the earth has been spread all over the world.