The idea is:
a.1) For each Daily Update, determine the total number of potential immigrants the player is entitled to
a.2) Show the player the a list of all potential immigrant’s and their (mind, body, spirit) statistics, so that the player can decide whether to accept or decline each individual immigrant based on their statistics.
a.3) If declined, an immigrant is gone forever (they don’t get re-offered at the next Daily Update).
b) Use a completely separate pseudo-random number generator for generating potential immigrants (their mind, body, spirit statistics; their appearance; etc) and nothing else.
c) Save the current seed for the “immigrant only pseudo-random number generator” in save games.
By saving the “immigrant only pseudo-random number generator” seed in save games it makes it impossible for the player to cheat (exit the game and reload repeatedly, until they get an immigrant they like); because that pseudo-random number generator’s seed will always be exactly the same after loading a save game.
By showing the player a list of potential immigrant’s statistics they have to make choice - accept a “less ideal” immigrant now, or decline the potential immigrant in the hope that they’ll be offered a “more ideal” immigrant later. Currently there’s no thought or skill involved in making a wise choice, because it’s a blind choice (you have no idea what you’re accepting or declining so there’s rarely any foreseeable reason to decline).
If the player keeps declining potential immigrants the number of potential immigrants they’re entitled to (and offered during the Daily Update) will increase over time (as their kingdom improves); so the longer the player waits for “more ideal” immigrants the more chance they have of finding what they’re looking for. However; various things already in the game (enemies, quests, etc) will naturally make this a risky strategy - if you’re too fussy and too patient, you’re not going to have well trained military, well equipped military, defences (turrets, walls, trenches) that are needed to handle the harder enemies that come with increased kingdom worth.