As much as you might believe you're entitled to a job, it never hurts to say "thank you" to your boss for having such confidence in your skills and capabilities to give you that promotion. Expressing appreciation for your boss's interest in your professional development is the least you can do, aside from actually exceeding her expectations in your new job.
A thank-you that expresses your sincere appreciation is always in order, regardless of how much you believe you deserved the promotion. A handwritten note is the best way to convey your appreciation in a professional manner with a personal touch. A well-written note goes beyond a simple thank-you email and it's often believed to be more of a professional courtesy than a routine email. If you have any doubt about whether to send a personal note or an email, write a note that you can personally deliver to your boss.
Refrain from giving your boss any kind of token of appreciation that might be construed as a bribe, no matter how ridiculous it might sound for you to be bribing your boss. You never know what others perceive as questionable or over-the-top when it concerns managers and employees exchanging gifts. Besides, if you and your boss have strictly a professional relationship and not a personal friendship, handing her a gift could make her uncomfortable and the truly awkward moment would be if she returns it to you.
If you're one of those employees who is fortunate to have a meaningful relationship with your boss on both professional and personal levels, meet privately with him to express your appreciation. Talk about how the promotion fits into your career plan and what your professional goals are from this point forward. If you're in a higher level job but in the same department, assure your boss that you'll always strive for excellence in your new job and that you appreciate his confidence in your aptitude to perform well.
When you're out to lunch with your boss, it's usually the highest-ranking person who typically picks up the tab. Although giving your boss a token of your appreciation -- such as a small gift -- can be awkward, it's perfectly acceptable to say, "I'd like to take you to lunch, since you've been so supportive of my career endeavors." Treating your boss to lunch also gives you an opportunity to discuss the next steps in your new job, reassure her that she made the right decision in selecting you for the promotion and say, "thank you" in a way that you may not be able to fully convey in a thank-you note.