It is broadly accepted that the Requests for Adminship on the English Wikipedia is not working well enough. Most of the proposals I've seen so far involve either (a) restricting who can participate in the process or (b) trying to place restrictions on how people participate (usually in order to avoid the process turning into voting). Solutions in the (a) group raise (wholly reasonable) objections of cabalism; those in the (b) group are doomed to failure because they will eventually mutate back into voting.
I think the real problem, however, that everyone is trying too hard to protect adminship. What if, instead of treating adminship as such a big deal, Wikipedia actually embraced its oft-repeated claim that "adminship is no big deal" and treated it as such? In addition to simplifying and detoxifying the adminship process (by making it much lower stress), it will help to undermine the appearance of adminship as a high-status position. If virtually everyone is, or can be, an admin, then the prestige associated with the position is diminished. And since all administrative actions are now reversible, the risk is a lot lower.
So, instead of doing things to make getting adminship harder, I instead propose to make getting adminship easier -- but impermanently.
The basic proposal: Any editor with one month's experience (that is, one month during which the editor has been consistently active) may request adminship. If nobody objects to the application during a five-day period, that editor shall be sysoped. If, however, anyone objects, then a discussion, not to last more than seven days, shall be initiated. Unless a consensus is formed why this editor shall not be an admin, at the conclusion of the discussion that editor shall again be sysoped. That is, consensus not to promote is required; it is not enough to have a lack of consensus to promote. Candidates for whom there is consensus not to promote must wait one month before trying again (and should in that time seek to remedy the defect that were identified in the discussion, if possible).
All administrators serve for a fixed three month term. After those three month, the editor is desysoped, and must sit out a month as a non-sysop before he or she may reapply using the above process. (This would not apply to admins explicitly exempted from this process by the Foundation, who would remain admins for as long as the Foundation says so.)
Finally, one final check: you can only object to one candidacy every three months. That is, if you object to someone's candidacy, thereby forcing a discussion, you can't object to anyone else's for three months.
My thanks to Triona, who discussed this idea with me, and offered several suggestions, some of which made it into this proposal.