Phoenix
The tradition that a phoenix has to fly overhead at the time of birth in order for a person to truly be a Phoenix might provide a way into the House... if the applicant could prove it.
Issola
Perhaps you have to make a Lyorn and a Jhereg shake hands and call each other brother... Or some other incredibly difficult demonstration of grace, tact, or diplomacy.
Tsalmoth
"Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "No.", "Can I please join your house?" "Oh, okay, fine."
Vallista
This house might have some kind of artistic test. (Presumably more elaborate than those found on matchbook covers.)
Interestingly, artistry-based admission implies that House Vallista could potentially admit Serioli members, if any House would. A race that invented Morganti weapons would surely fit their creation/destruction motif, after all.
Chreotha
Maybe you have to trap and hold a Chreotha Council member for seventeen days?
Yendi
Yendi might let into their House someone who was clever enough to con, bamboozle and manipulate 17 Yendi.
--Actually, if someone is twisted and manipulative enough to enter into the Yendi House, they would do so without anybody else knowing about it. Therefore, actually letting people in is irrelevant. If someone flat out asks to join, they're too blunt to be a Yendi.
Orca
Those who can prove great deeds as (probably nautical) explorers may be able to petition for membership in the Orca. Considering the House's preoccupation with wealth, the Orca might also be the only House other than the Jhereg that one can buy one's way into.
Athyra
The House of the Athyra might admit an exceptionally talented sorcerer.