You should begin thinking about your thesis as soon as possible. Many students find the idea daunting, but the more thought you give to this early, the easier it will become.
To begin the process, you should identify an area in which you are interested and with which you want to spend several semesters working. You should then contact the appropriate faculty member and ask him/her to serve as your advisor. Presumably, this person will be someone with whom you have had a class or two, with whom you know you will work well.
Working together with your advisor, you will define the nature and scope of the project. Consider applying for a UCARE grant to support your work.
Do not hesitate to contact Dr. Berger (email@example.com) or Dr. Lyons (firstname.lastname@example.org) for assistance in this area. A thesis handbook is available through the main Honors office. Dr. Lyons offers an honors thesis seminar (see the 198H Seminars). Be sure to submit your thesis completion document by the deadline (see the Deadlines section).
Some colleges have specific requirements and procedures to use a research project to address "graduation with distinction" issues. Know whether your college has such a policy by visiting your college's undergraduate advising office or the Honors Office for information.