Instead of choosing between simple selections or "Random", provide multiply selectable options.  That is, instead of having to choose between a single fixed option or a random choice that draws from ALL possible choices, provide multi-selections for those options from which a Random choice can be determined.


More specifically, have it so each of the possible Galaxy Shapes can be checked or not checked; if more than one is selected when hitting Start, the generator can then choose randomly between those selected options.  The same can be done for all the game generation options:  player's chosen empire, AI empires (individually or as a pool), galaxy density, galaxy size, game difficulty, number of competitors, game speed, etc.


This would provide more focused random game generation: I could start a game where the player empire is randomly selected between three favored empires, generate a galaxy randomly from one of four favored shapes, with the galaxy density either medium or high, but not low (two of three selected), with the game speed fixed at fast (only one selected), and so on.


Do to the same now, I would have to set everything as random (that can be randomized) and then reject those games generated with the options I did not want.


As a corollary, all the game options without a random option should instead include the benefit of random selection:  pirates (yes, no, or randomly either), minor civ difficulty (easy, medium, hard or randomly from one of four combinations), empire spawn method ... well, you get the idea.