Using the Eldritch Adept feat, a non-warlock can take any invocation that doesn't have a prerequisite. One invocation which meets that criteria is the Thief of Five Fates, which states:
You can cast bane once using a warlock spell slot. You can’t do so again until you finish a long rest.
Is this a worthless invocation choice for someone without Warlock spell slots or is there some way which they could utilize this casting of bane?
The text of the Eldritch Invocation is pretty clear...you can cast bane using a Warlock Spell Slot.
If you don't have Warlock Spell Slots, you cannot cast it. This same rule applies to a multi-classed Warlock: they have to use a Warlock spell slot for this, they can't use a normal spell slot that they acquired from, say, Multi-classing Sorcerer.
The language used across these Eldritch Invocations is very consistent. If it was intended that you could cast these spells using some other Spell Slot, then it wouldn't specify that you had to use a Warlock Spell Slot. It would instead say something like...
You can cast bane once using a spell slot.
But it doesn't. It would have been very easy for them to leave the specificity out, and they did not. Thus it's a safe bet that this is intentional.
This is supported by an interview with Jeremy Crawford (lead rules designer for WotC), found here. To transcribe...
Bart Carrol: "So, for example, if it requires a warlock spell slot, then in that case it would require a warlock to be able to use it?"
J. Crawford: "What this feat is saying, is that it doesn't matter what the prerequisite is. If an Invocation has a prerequisite of any kind, only a Warlock can take it"
While not a definitive ruling (as it's just an interview, not a formal Sage Advice post), and is using Crawford's usual circuitious language...I find this to be fairly strong support. Crawford appears to consider "you need a Warlock Spell Slot to cast this" to be a prerequisite.
Features and abilities do what they say they do and nothing else. Since the invocation doesn't add Bane to your list of spells known nor does it grant you a spell slot, you have no means to cast it.
It's also important to note that Warlock spell slots from the Pact Magic feature are treated differently from spell slots gained via the Spellcasting class feature when multiclassing. Technically speaking, you can't just use any old spell slot to cast the spell, it specifically must be a spell slot gained from levels in the Warlock class.
All of that being said, were I a DM in this situation, I would allow a player to cast the spell via any of their available spell slots, after all, we should generally lean towards more permissive rather than less permissive. That being said, you'd still need an actual spell slot though. If you wanted to be able to cast it without a spell slot, then I would recommend you take the Magic Initiate feat.
Under one interpretation of the rules as written, no feature besides spellcasting from a non-warlock class can be fueled with warlock spell slots, and no warlock feature besides pact magic can be fueled by non-warlock spell slots.
The multiclassing rules for spellcasting tell you to add up your spellcasting slot levels from multiple classes, except for warlock pact magic. It then explicitly lets you use warlock spell slots to cast non-warlock spells, and non-warlocks spell slots to cast warlock spells.
No text permits you to use warlock spell slots on non-warlock class features, and vice versa.
Under this interpretation, you cannot smite using warlock spell slots, you cannot use flexible casting using warlock spell slots, and you do not regain your non-warlock spell slots when you take a short rest.
The alternative, allowing warlock and non-warlock spell slots to be freely used whenever the rules say "spell slots", leads to madness in a few ways.
First, and most famous, is the coffeelock. This uses flexible casting from the sorcerer to sacrifice short rest spell slots for sorcery points, spend them to gain non-warlock spell slots that last until a long rest, and take short rests to regain your warlock spell slots.
A second and even more abusive is the wording of pact magic, which states:
You regain all expended spell slots when you finish a short or long rest.
without the "features always refer only to the spell slots gained by this class" (and the ones combined from other classes), this literally means a wizard 11/warlock 1 regains all expended spell slots from both wizard and warlock when they finish a short rest.
All of this means that the use of "warlock spell slots" in the invocations doesn't matter.
Now, as a DM, you should neither use the "slots are slots" or the "slots can never be used by other class features", but rather deal with it on a case by case basis.
Here bane is a level 1 spell. Permitting you to cast it using any spell slot 1/rest is not going to cause a problem.
Other cases, like polymorph, you might require that it be a 4th level slot or better for non-warlock slots.
Similarly, permitting warlock slots to fuel smite isn't an issue. Coffeelock is.
