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In Ambition and Free modes, it's primarily affected by the difficulty, and Blacksmith's inventory in Ambition stocks better weapons once you upgrade the facility.
You can also equip and level the Fortune Hunter skill to increase the odds of gettting better elements or unique weapons to drop.
The absolute best weapons, i.e. 5- and 6-stars, however, are unlocked by fulfilling specific objective on a specific stage (Story or Free mode) with a specific character on Hard difficulty or above. Once you've beaten the Story, you can check the conditions in the Gallery under "Weapons".
Character level doesn't factor into the odds, as far as I know.