The "free" in "free object interaction" means without cost – the object interactions you are permitted to have without having to spend your action on "Use an Object". In "Other Activity on Your Turn" (PHB190)
Your turn can include a variety of flourishes that require neither your action nor your move…. You can also interact with one object or feature of the environment for free, during either your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack. (emphasis mine)
And in "Interacting with Objects Around You" (ibid)
Here are a few examples of the sorts of thing you can do in tandem with your movement and action: (emphasis mine)
My question here is Rules as Intended, how much emphasis to place on the phrases "as part of" and "in tandem with"?
It could be that you are allowed to interact with one object regardless of your action, and anything of roughly the complexity of everything listed under the second citation above is fair game, regardless of whether it thematically fits your main action.
Or, it could be that your free object interaction can only be used if it "fits" as part of your main action.
So, for example, if I want to draw a sword and attack with that sword, that clearly is part of my attack action and would be allowed under either interpretation.
Likewise, if I wanted to pull out a focus to use in spellcasting, that would be allowed as part of my cast a spell action.
But what if I began my turn with a sword in hand and wanted to put it away so that I had the hand free to perform the somatic component of a spell?
A generous interpretation of the free object interaction would say that "sheathing a sword" is specifically on the list of free interactions, so it is allowed as part of any action.
A more restrictive interpretation would say that sheathing a sword is not, strictly speaking, something that could be considered part of the casting a spell action so that while I could drop the sword (as when I change weapons before an attack), I could not carefully sheathe it.
Aside from the actual PHB rules (which I consider vague), the only guide to interpretation I have found comes from Sage Advice of 3/2015:
[A cleric] likes to wade into melee combat with a mace in one hand and a shield in the other…If [she] casts cure wounds, she needs to put the mace or the shield away, because that spell doesn’t have a material component but does have a somatic component. She’s going to need a free hand to make the spell’s gestures. If she had the War Caster feat, she could ignore this restriction.
Unfortunately "put the mace or shield away" doesn’t provide a lot of guidance on whether this means carefully stowing as part of her cast a spell action.
A good answer will not just rehash the cites I gave above and argue for one specific interpretation of the words over another. Rather, it will find other citations, either in official publications or designer intent tweets and such, and explain how they specifically favor one interpretation over the other.
Related: Drawing and sheathing weapons as one item interaction