General Considerations
Most non-profits either have self-perpetuating boards of directors, or have members or delegates whose elect a board. Membership in a typical non-profit would be defined to include donors and/or volunteers, or people who have affirmed the organization's principles. In very large organizations, members frequently elect delegates to a convention or annual meeting and the delegates rather than the members directly elect the board of directors. But, there are non-profit corporation statutes that are better suited to these kinds of structures than an LLC.
A non-profit LLC can have members and/or a manager, in lieu of a board of directors. But, the only good way to convey how an LLC could be used as a non-profit is really by example. Almost all of these LLCs would have very custom terms - you couldn't use an off the shelf operating agreement in many of these cases.
Being member operated and controlled is just fine (although a manager managed form of LLC almost always works better in my experience), but being member "owned" is only O.K. in a quite restrictive sense of the word "owned" because basically, the whole point of a 501(c)(3) is to have an organization that is "unowned" where there is no one who can privately benefit from it in the crass economic sense.
While an LLC can be a good fit for some niche applications of non-profits, it is an organizational form that isn't a good choice for all non-profits and only makes sense where it addresses issues that a conventional non-profit corporation or unincorporated association does not. As a default for a "generic" non-profit, a non-profit corporation or unincorporated association should be the first choice and the LLC should be used only when it provides a more natural fit to the organization, which will be more than exception than the rule.
Unlike business organizations, which mostly have a great deal in common with each other, non-profits have very diverse governance needs that can't be applied generically and instead their governance must reflect the substance and details of what they are trying to do.
A core principal of a 501(c)(3) (although not all non-profits) is a lack of personal benefit to the people who control it.
Examples
There are various ways that LLCs could be used.
One is as a joint venture of multiple non-profit members. One might, for example, have an LLC whose members were the various synagogues of a region for the purpose of jointly operating a regional Jewish Community Center.
Similarly, LLCs can make more sense for non-profits that are not 501(c)(3)s in many instances. It could be a good organizational form for organizing membership in a country club or condominium association or stock exchange, where transferability of a more or less permanent membership would be a useful feature.
In a 501(c)(3) with a permanent transferable membership, you have a couple of choices.
One would be to give members voting rights but not economic rights, which might make sense, for example, for an organization that conducts an periodic family reunion or governs a family charitable foundation, or perhaps some other hereditary type non-profit like the Daughters of the American Revolution. One could similarly imagine an LLC organization for a non-profit compose of property owners or de facto trust protectors in an LLC whose purpose was to enforce a conservation easement or a covenant that property in a neighborhood or HOA was maintains as open space.
Or, for example, perhaps membership in an LLC running a 501(c)(3) library of ancient books or museum with art or ancient relics could be attached ex officio to endowed professorships in appropriate departments at universities across the nation or the world, or to people holding clergy positions - for example, memberships might be ex officio to specific bishop or archbishop positions in one or more church denominations (this might work well, for example, for a joint venture of multiple Orthodox denominations - Greek, Russian, Ethiopian - that want to cooperate with each other).
Not all non-profits are permanent either. An LLC could be a good model for a single project, limited time period non-profit venture. Donors could have voting memberships equal to their contribution, while the promoter of the idea of a kickstarter-like project like a non-profit documentary project, or a one time event, would be the manager of the LLC but could be removed and replaced by a majority vote of the members if the manger floundered or quit or engaged in misconduct. Any funds left over after the event would be donated to a pre-agreed (or an after the fact member determined) charity.
Another approach would be to flip the script even more completely where members were required to make annual contributions rather than receiving annual profits, and could relieve themselves of this obligation only by transferring it to someone who assumes this obligation. There are synagogues that are functionally organized somewhat like this and this also might be a decent fit for a cemetery association. Japanese cemetery associations are organized in a manner somewhat similar to this kind of structure. More grandly, the United Nations and many of its associated institutions are organized somewhat along these lines.
Yet another format would be to have a single member LLC form on the model of a "corporation sole" which is how most Roman Catholic institutions and many other religious institutions are organized. This is essentially the same idea as a trusteeship with some succession mechanism worked out. This could also be a pretty good fit to the traditional organizational structure of many mosques, which are often established primarily by a lead donor and his family in a manner a bit analogous to a private foundation in the West.
