As someone who has had a Bar Mitzvah service and officiated other people's B'nei Mitzvot services, that is an interesting question and easy to answer. What makes this Jewish tradition such an effective coming of age ritual?
(I will only discuss the traditional elements, not the very recent trend to emphasize a party after the religious service.)
Receive a Related Pair of Adult Responsibility and Privilege
It is sensible that a coming of age ceremony is when the new adult first receives some of the responsibilities and privileges that adults share but children do not have.
The phrase "Bar Mitzvah" literally means "Son of the Commandment". People become a Bar Mitzvah, not have one. (Similarly "Bat" means "Daughter" and "B'nei" means "Children".) In Jewish tradition this ceremony marks when a new adult becomes responsible for his own morality and behavior, departing from being under his parents' responsibility. The corresponding privilege is that the new adult may now participate in adult discussions and judgments about morality. (For example, the how do the ancient Jewish precedents about gossip and slander apply to when congregants write falsely and maliciously on Facebook?)
This is also easy to translate into a secular coming of age ceremony. Find a related pair of a responsibility and a privilege with which the new adult can participate.
Lead an Adult Community Activity
The core of becoming a Bar Mitzvah is that the new adult takes a turn leading the weekly religious service. Traditionally, the new adult is responsible for all of the rabbi's normal roles: directing the congregation in prayer, calling up the other people who have turns participating, reading from scripture, and presenting a sermon. (The cantor still helps lead the singing.)
This is probably the first time the new adult has complete responsibility for a meeting or event. It is the new adult's fault if things fall flat.
This is easy to translate into a secular coming of age ceremony. Lead a community activity.
Learn from a Non-Parental Authority Figure
For months prior to the ceremony the new adult studied under the rabbi. These days a one-on-one mentoring relationship may not be so significant. Historically, before communal schooling of children, this could be the new adult's first structured learning under a non-parental authority figure. As a child, the Bar Mitzvah would have attended many public discussions of scripture and religious issues. But this would probably be the first one-on-one training outside the home.
Part of becoming an adult is identifying non-parental authority figures worth following. Also, a new adult who proved teachable by the rabbi could be viewed as teachable by master craftsmen looking for apprentices.
This aspect of a coming of age ceremony is difficult to translate into secular culture. So many American subcultures have authority figures that are not mentors but idols.
Perhaps the only useful generalization is an essay question: Reflect about which adults besides your parents you look to as respectable authority figures. Why those adults? What teaching do they provide? How do they guide you? What does all this say about you?
During the ceremony the most stressful part is usually presenting the sermon. This is probably the first time the new adult has taught a room full of adults.
This is also easy to translate into a secular coming of age ceremony. Teach a gathering of adults something.
Begin Cooperatively Process Group History
Children are often taught "Our family does this..." or "Our people value this..." A coming of age ceremony is a time to discuss your family's historical narrative: what your ancestors have experienced and what path is most open before you. Which elements of group identity and destiny is the new adult free to accept and reject? Which are unavoidable?
Similarly, it is a time to ponder which group beliefs are time-tested values, which are dysfunctional historical relics or prejudices, and which are the idiosyncrasies of parents or community leaders that have added flavor and perhaps joys or aggravations to childhood years.
Older children do think about these issues. But they are seldom provided with structured time over several months to really discuss them with a non-parental adult who is judged by the community as insightful and worthy of respect.
In Jewish culture this discussion happens extensively during the months before the ceremony when the new adult studies and talks with the rabbi. But the issues are the same in all subcultures: the only difficulty in translate into a secular coming of age ceremony is identifying an appropriate non-parental adult.
How do we translate this into a secular coming of age ceremony? Discuss how you fit into the history, values, and world-views of the group(s) to which you belong. Learn enough to represent those groups with explanation, even in those aspects you choose not to represent in habit and deed.
Begin to Earn Adult Respect
One problem with being an American teenager is that society does not give you ways to earn esteem. The ways children earn esteem no longer apply: sharing, playing nice, trying hard, and being open to new experiences are now assumptions rather than rewarded behaviors. The ways adults earn esteem probably do not apply: outside of very agrarian subcultures most teenagers simply cannot do significant work to help support their families, and in recent generations most families have lost the ways to contribute economically at home by using dried beans, canning and dehydrating, etc. Teenagers are given only unreasonable sources of esteem: be a straight A student, be a sports star, etc. No wonder teenagers get in trouble for inventing their own sources of esteem and pursuing these unhealthily!
There is another nearly universal source of adult esteem that is often overlooked because most American adults fail to acquire it: participating well in conversations about religion, politics, and culture. Most adults simply take turns sharing their own observations or opinions rather than building off what others have said to create a genuine conversation. How many people do you know whose contributions in discussions about those topics routine add clarity or insight to the conversation?
I have already mentioned sermon preparation and reflection upon group history, identity, and values. One goal of that preparation for the coming of age ritual is to make the new adult someone that does talk well about religion, politics, and culture. As a new adult this ability might not have much breadth: the number of topics for which the person can contribute with clarity and insight will probably be small. But the skills of effective participation and contribution are widely applicable.
This is also easy to translate into a secular coming of age ceremony. Be mentored in the knowledge and conversational skills shared by people who others enjoy having lunch with while talking about religion, politics, and culture.
Together all of these provide a chance to demonstrate why other adults should now pay more attention to the new adult. Instead of a child who runs and plays at the periphery of the community, the new adult is a central member of the community. Does it matter?
This is the new adult's chance to say, "Look at me. See that I can handle adult responsibility. Hear what I have to contribute to adult conversations. Sample why I am pleasant to spend time with as an adult."