Have them be a spellcaster with access to high level spells

The easiest way to achieve effective immortality is to use spells. A wizard has access to several of them once they get higher level spell slots.

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  Sequester will allow the BBEG to put themselves into a state of suspended animation during which they will not age. This will allow them to span any amount of time without aging (though it will mean that they are out of action for that time).


• 
  Clone will provide the BBEG with fresh young bodies to use once their older body dies. With enough of these, they can last an infinite amount of time. See this Q&A for more discussion on this.


• 
  Wish obviously wish can do anything, including granting immortality. A wish for immortality is unlikely to be granted in a straightforward manner though and would be a good opportunity for the DM to implement some sort of curse or other downside on the BBEG as a result of their wish.


• 
  Imprisonment is an option but it would require someone else to cast it on BBEG and for them to let them out again.


• 
  Reincarnate, revivify, resurrection, true resurrection can obviously also bring BBEG back from death assuming there is another caster capable and willing to cast it on them.

Instead of having the BBEG be a spellcaster, you can always have them have the assistance of someone or something capable of casting the spells. For example, Djinni can cast wish as well as some magic items. This can be a good solution in the case that making the BBEG high enough level to gain access to the spell themselves would be too much for the party to fight against.

Be a high level Druid or Paladin

At 18th level, druids get the Timeless Body feature:

[T]he primal magic that you wield causes you to age more slowly. For every 10 years that pass, your body ages only 1 year.

Combine this with a long-lived race like the elf or warforged and you easily have thousands of years of life.

An Oath of Ancients paladin at level 15 gets:

Additionally, you suffer none of the drawbacks of old age, and you
can’t be aged magically.

Which seems likely to be able to prevent death from old age as well if your DM interprets death from old age as a drawback. (thanks @Walt)

Grant an Epic Boon

As Carcer points out in their answer, a DM can use the Epic Boon of Immortality to grant BBEG Immortality. This assumes that they have 20 PC class levels and may not be justification enough for BBEG being immortal plot-wise, but it is an available mechanic.

Magic items

The Ring of Winter listed in Tomb of Annihilation will prevent aging.

You, as DM can also feel free to invent any kind of item or artifact that can prolong life as well. Just be sure you are fine with having it fall into the PC's hands or have some sort of plan to have them need to destroy it.

In fact...

Make something up

If none of the RAW options above satisfy you, make something up. You are the DM and you have the power to change your world to contain whatever effects you want. Perhaps, the villain found or made a powerful artifact. Or maybe they found a sacred site or ritual that allows them to be long-lived. The options are really endless and really, there shouldn't be much of a concern here because the long-lived villain is a common trope and is not likely to be a problem for players to accept even without explanation.

Or even better, you can mix and match from all these options to create the long-lived villain of your dreams.

The Epic Boon of Immortality

The DMG (on page 231) describes an optional system of further advancement for characters at 20th level called epic boons:

An epic boon is a special power available only to 20th level characters. Characters at that level gain such boons only if you want them to and only when you feel it's appropriate.

[...]

Epic boons can also be used as a form of advancement, a way to provide greater power to characters who have no more levels to gain. With this approach, consider awarding one epic boon to each character for every 30,000 XP he or she earns above 355,000 XP.

One of the boons described on page 232 is the Boon of Immortality:

You stop aging. You are immune to any effect that would age you, and you can't die from old age.

These options are intended for player characters, but there is no reason you cannot also use them for your NPCs. This method does presume your NPC is a 20th level character; that could be a challenge for a ~15th level party, especially if the BBEG is of a class which gets a very strong capstone ability, like a Circle of the Moon Druid, but it's not a totally insurmountable difference, depending on the circumstances of the fight.

MacGuffin to the rescue

If you're the DM and you want the BBEG to have lived for thousands of years, then he's lived for thousands of years. Literally just make up a reason (try to make it a good reason); demonic pact, other worldly blessing, unicorn blood, whatever.

You don't need any mechanics to support you. Your rule is law.

If you really feel like you need a pseudo mechanical reason, then there are enough weird class features (Timeless Body for an 18th level Druid) or spell shenanigans (clone cycling) to keep your BBEG kicking for millennia.

Use a monster

I know you said you didn't want to, but why not? Devils are immortal (as well as plenty of other things), evil, and intelligent. It seems to fit the bill, but what you want is what you want....

Which brings me to a subpoint. Just give them the stats of a devil of your choosing and then make them literally anything else. The DMG and MM even suggests this (citation needed).

So this sounds pretty dumb, but quite doable.

For something that appears to be immortal, but has a high reset cost, do glyph of warding -> true resurrection.

In order to cast true resurrection out of glyph of warding, glyph of warding itself must also be cast at ninth level during setup. This means in turn that at least one of these spells has to be cast from a scroll due to both demanding a ninth-level spell slot.

So you have your BBEG hire* a 17th-level cleric to have a bunch of scrolls of true resurrection made. By RAW, scrolls of true resurrection take quite a long time to make, but that all took place before your campaign even started, so what's the problem?

^{*insert preferred manner of persuasion here}

I've once used a concept to create a "eternally changing" BBEG, but you can use it to create a effectively ageless villain. She is inspired by a story I read quite a while ago and I unfortunately can't remember the name nor the author.

Her gist was that she stole other people's bodies. Every once in a while, she would task her cult to bring her a few "candidates" from their tribe. Those candidates, from both genders and all walks of life, would then compete on a series of tasks and one of them would ultimately be chosen to replace the now-aging regent as their cult leader.

Once the next "ruler" was chosen, this person would enter a special chamber where the current regent would supposedly perform a ritual to pass on their knowledge to the next regent and then pass away.

However, what the ritual actually did was to swap the souls of the regent and the candidate, thus leaving the big bad in a new, different and young body. The unfortunate victim, now in an old and decrepit body, would then be pushed into a giant chasm directly to a pit of lava, erasing any hint of what actually happened in that chamber.

The "new" leader would then appear towards their people once more, greeting everyone with a very distinct red tint on their eyes - the mark of "The One" - and hold a feast to celebrate the regent that passed away and the one that now rises to power.

And everybody would go along, happy as if everything was good and right.

In terms of rules, this permanent "body swap" would be done via a Wish spell.