OK. Lets start throwing things out there.
I have suggested levels for these, but could do with suggestions to fit them better.
These are mostly based off the concept of the Fighter as a highly trained and practical person with solid knowledge of tools and items other than weapons.
Nothing here should be of excessive power or step on too many toes, particularly given most fighter's stat distribution: They are designed to give the fighter some non-combat options to aid the party narrative without impinging on the power of most spells or rituals.
Grounded: A fighter's training is generally less esoteric than a spellcaster, and less isolated than a Rogue's. It often brings them in contact with the lives of people around them.
The Fighter gains an additional Background, with associated proficiencies.
At 6th level, the Fighter may apply their proficiency bonus to Intelligence, Wisdom, or Charisma checks relating to people within either of their backgrounds. If they have the relevant skill proficiency, they may apply twice their proficiency bonus.
The Fighter's experience with tools also extends beyond those used for killing. At 6th level they gain proficiency in two sets of Artisan's tools, or vehicles. If they are already proficient, they may apply twice their proficiency bonus to checks involving them.
Practical planning: The Fighter is an old hand at packing and carrying gear, planning ahead and improvising items. Once per long rest, the fighter may produce mundane items that they had not declared on their character sheet. The character must lose funds equal to the item's price when they produce it, and the items cannot weigh more than the character's Strength score. If the item could be made by a set of tools that the character is proficient in and has available, this weight limit is doubled.
Strife in all its forms: The Fighter can apply their training and knowledge of combat to noticing related details, and their perception and common sense to issues of strategy and tactics.
When making an Intelligence or Wisdom ability check relating to combat, the Fighter may apply both their Intelligence and Wisdom modifiers to the roll. Combat-related checks could range from Wisdom(Insight) checks to determine whether a thug is ready to fight or just posturing, to Knowledge(History) checks to recall details about a historic battle. The DM is the arbitrator of what constitutes combat-related.
The Fighter also adds both Intelligence and Wisdom bonuses to their Initiative.