http://chareidi.org/archives5761/korach/KRfeatures.htm
While our rabbis tell us that davening at kivrei tzaddikim is a
segulah for all types of help, the tradition of Amuka as an address
for unmarried men and women is a relatively new one, according to
experts in the field. It was "rediscovered" about fifty years ago by
Rav Shalom Gefner of Meah Shearim, among many other holy grave sites,
which he ferreted out according to the writings of Rav Chaim Vital,
based on the teachings of the Arizal.
A well-known famous Tsefati, Hershel the Shamash and Rav Berel Tsfaser
knew the location of the tsiyun of Yonoson ben Uziel and told Rav
Shalom about it in 1951. There were no roads in the area at that time.
Reb Shalom began to come to the tsiyun with groups of young men:
braving the elements, traversing dirt paths and scurrying down rocky
mountainsides in order to reach it. Some of these adventurous bochurim
were of marriageable age, and some became engaged soon after their
trek to Amuka, the story goes, for lifum tsa'aro agro. . . And more
bochurim made the trip, and more eligible young ladies. . .as roads
were paved and access was facilitated.
In a book called, Tovah Ri'iyata from 5734 (1974), it is said in the
name of Rav Chaim Lichtenstein (who was niftar in 5725) that, "Prayer
at the tsiyun, of Yonoson Ben Uziel is a segulah for shidduchim." This
is the earliest written mention of the place being a segulah for
shidduchim.
Explanations have been offered ex post facto to explain the segulah.
Some say that since Rav Yonoson never married, he helps others find
their true mates as a kaporoh, but there seems to be no reason to
believe that he in fact remained single.
Some, probably as a joke, use a source in Rashi on Yevomos 17a: "All
pisulim that don't find a wife go there." The next Rashi begins "Vehi
amuka," referring to the next statement in the gemora.