Claudia is not running away from home for good.  The whole purpose of running away from home is to show her parents how important she is.  Claudia is the oldest, and she is feeling unappreciated.  Her life is boring, and she always has to do the chores while her brothers don’t.  This is why she saves up her money to run away.  It will accomplish both purposes.  Her parents will realize how valuable she is, and she will get some excitement.

Since she intended to return home after everyone had learned a lesson in Claudia appreciation, she had to save money for her return trip, too, which was like full fare one way. (Ch. 1)

Claudia wants her parents to know what happened, so she mails her parents a letter, telling them she ran away and not to contact the FBI. She doesn't seem to think they will panic.  That is still probably more practical than sending the letter with the Corn Flakes box tops to get twenty-five cents.  Yes, she may need the money, but how does she expect to get it?  Jamie points this out.

Claudia chose Jamie to accompany her in running away mostly because he had money.  Even though he was nine, he earned money cheating at cards, and he also saved his money.  Claudia had been saving for three weeks and still did not have as much money as he did.  He wasn't thrilled with her idea to run away to the Metropolitan Museum of Art in New York City at first, but she was older.