I decided that GNU octave would be a nice place to start. If so, please advise me on how to proceed and let me know how I may contact you for additional assistance. I'll be free in October because I'm presently taking exams, so I'll be able to start then.
GSOC is a program managed by Google usually run over the months of May-August each year. The organizations apply for their own inclusion at the start of the year, then students apply for selection around March. So I'm not sure what exactly it is you're looking for at this time regarding the ability to start something in October.
It is always strongly encouraged for prospective participants to familiarize themselves with, and even begin contributing to, a project before the application process. If this is what you mean, you could start looking through the Octave wiki [1,2] and outstanding bug list [3] for ways to get started contributing. Obviously Octave maintainers could help with questions and guidance about particular contribution issues, but you don't need their permission to get started.
Also, much Octave discussion has moved from these making lists to https://octave.discourse.group , which would probably be a better place for any followup questions.