What are the most popular gender-neutral pronouns that aren't the same as other contemporary pronouns?

Ze and hir have been around for some time, although they haven't gained much traction in the cis community. There're more listed here: http://en.wikipedia.org/wiki/Gender-neutral_pronoun#Modern_solutions

There are no such pronouns in common use in any major dialect of English, though that's not for lack of trying. Since at least the 1850s, various linguistic do-gooders have proposed more than 80 potential gender-neutral pronouns, none of which ever gained traction. A sampling:

thon, thons; ne, nis, nim; hi, hes, hem; e, es, em; ir, iro, im; ip, ips; he'er, him'er, his'er, his'er's; te, tes, tir; shis, shims, shim, shimself; zie , zees, zim, zeeself; per, pers

All of these have the significant drawback of being ridiculous, which is probably the reason singular they is the only option that's come close to mainstream acceptance. Until the prescriptivists accept singular they, we're stuck with rewriting to eliminate the gender-specific pronoun.