Simple possible answer

They may have picked it up from the Scotch-Irish settlers of Appalachia. These early residents left many vestiges in the language spoken there.

Notice how close that pronunciation is to some of those from the north of Britain or from Ireland as shown below under Technical Details.

---

If you mean monophthong /e/ without the offglide here that many speakers have but which is otherwise indistinguishable from the diphthong you're used to, this is perfectly common albeit not universal in North America.

We're somewhat famous for our monophthong /e/ and /o/ phonemes here, both tense not lax. This will likely be the same tense vowel you hear us say in ate, hang, reign, beige, face, main, gauge, bury, very, or in the name of the letter H. (Given that this also happens in Scotland, you might not want to be too quick to blame it on our own Scandinavian ancestors, or even on theirs.)

However, if instead you mean the same lax phoneme /ɛ/ that many folks have in most and often all of many, said, Craig, phlegm, bread, men, catch, leg, dress, then that is something different.

Along these same lines, you yourself might even have this lax dress vowel in ate or in leisure, or both. But if you do, I have little doubt that those whom you find strange saying naked with a lax vowel would find it strange to hear people like you saying ate or leisure with that same lax dress vowel.

Others you might even hear saying bag with this lax dress vowel.

In all these cases, it's just how they talk in that part of the world, that's all. There wouldn't be any reason for it anyone could normally pinpoint. Nearly any English word will have dozens of normal pronunciations scattered not merely across the Isle of Britain but indeed all around the entire globe.

This is what happens when a language that has been spoken for many centuries comes to be spoken by a billion people separated sometimes by as much as half a world. It is completely inevitable under those circumstances.

These minor variations only sound strange the first time you hear them. But usually complete sets of words shift, split, or merge together in concert with each other. They'd probably all move at the same time.

You'll quickly get the hang of it anywhere you spend any substantial time among, and you'll stop noticing it as weird. It's just another way of talking, that's all.

---

Technical Details

On the UK Sound Comparisons website, they specifically demonstrate how naked is normally pronounced in forty different regions or cultures. If you go to their site through that link, they kindly provide actual audio for each of these.

I’ve annotated these locations as follows:

• Unmarked: an [eɪ] falling diphthong the way it is in RP, in New York City, and in Chicago, so it has a tense [e] as its syllabic nucleus followed by a very slight offglide.
  
  Tense vowels are also sometimes called close vowels, or even (mostly incorrectly) ‘long’ vowels.

• Italic: an [ɛɪ] falling diphthong, so lax [ɛ] as the syllabic nucleus followed by a very slight offglide. This is fairly close to the previous one.
  
  Lax vowels are sometimes called open vowels, checked vowels, or even (mostly incorrectly) ‘short’ vowels. These are probably a little bit like the dress sound that I’m guessing you’re surprised by, but they still have an offglide and so are still diphthongs. It’s just that they have an open nucleus not a close one.

• Bold: a tense [e] monophthong. This is fairly common in Britain, too, but it has the same tense vowel as RP, just without the offglide.

• Bold Italic: a lax [ɛ] monophthong. This is the dress vowel that I’m guessing is surprising you.

• A †dagger†: something else that isn’t one of the four listed above.

My hunch is that it’s the versions with just a lax monophthong that surprise you, such as from Alabama, but these also occur in the British Isles in places like Morley, Middlesbrough, Lindisfarne, and Donegal. Such places I’ve set in bold italic so you notice them.

The North Carolina diphthong has a lengthened lax [ɛ] before the glide in its falling diphthong, so that too might stand out to you. For the most part you should just ignore the little diacritics, but in this case I’ll fully decode those for you. That complete [ɛ̞ˑɪ] vowel there means:

 ɛ̞ˑ  open-mid front unrounded vowel          U+025B  LATIN SMALL LETTER OPEN E
     lowered                                 U+031E  COMBINING DOWN TACK BELOW
     half-long                               U+02D1  MODIFIER LETTER HALF TRIANGULAR COLON
 ɪ   near-close near-front unrounded vowel   U+026A  LATIN LETTER SMALL CAPITAL I

That’s quite close to the sound heard in Somerset, just a bit longer.

England (South, West Country, and Midlands) and Wales

1. Somerset [ˈnɛ̝ɪkɪd]
2. N. Devon [ˈne̞ˑkɪd]
3. S. Wales [ˈneˑkɛ̈d]
4. Received Pronunciation (RP) [ˈne̞ɪkɪd] If you speak Standard Southern British English, then you probably have this diphthong.

England (North)

5. Buxton [ˈnɛ̝ɪkɪd]
6. Morley [ˈnɛ̝ˑkʰɪd]
7. Middlesbrough [ˈnɛ̝ˑkɪd] This is pretty close to Alabama’s.
8. Cornhill [ˈneˑkɪ̠d]
9. Longtown [ˈne̞ˑkʰəd]
10. Lindisfarne [ˈnjɛkɪ̠d] This one is actually a rising diphthong.
11. Tyneside [ˈneˑkʔ͡əd]
12. Liverpool [ˈneɪkəð̠]
13. Rossendale [ˈne̞ːkɪd]

Scotland & Ireland

14. Scotland, Standard Scottish [ˈneˑkɪ̠d̥]
15. Scotland, Hawick [ˈneˑkɪ̠d]
16. Scotland, Coldstream [neˑkɪd]
17. Scotland, Edinburgh [ˈne̝kʰə̟ ̝d]
18. †‘Doric’ Scots: Buckie† [njɑ̈kʰə̝tʰ] Another rising diphthong, and somewhat unusual.
19. Scotland: Lewis [ˈne̝ˑkɪd̥]
20. Ireland, North: Antrim [ˈneˑkət]
21. Ireland, North: Belfast [ˈnëˑk̟əd]
22. Ireland, Tyrone [ˈnëˑk̟ət]
23. Ireland, Galway [ndeːkət]
24. Ireland, Donegal [nɛ̝kɪt] Except for the devoicing of the final consonant, this is pretty close to Alabama’s version. Do you think the Scots and the Irish also talk curiously in some places? Many of them settled America, notably including Appalachian states like you mention.

North America

25. North Carolina [ˈnɛ̞ˑɪkɪd], marked in bold italic because you may not hear the offglide after a lengthened nucleus
26. Standard Canadian [ˈnɛ̝ɪkɪd]
27. Standard American [ˈnɛ̝ɪkɪd]
28. Ohio [ˈne̞ɪkɪd]
29. Chicago [ˈne̞ɪkɪdʔ]
30. †Chicago AAVE† [ˈneikɪ̠t]
31. Boston [ˈneɪkɪd]
32. New York City [ˈne̞ɪkɪd̟̚]
33. North Carolina AAVE [ˈneɪkɪd]
34. Alabama [ˈnɛkɨ̞d]

Rest of the World

35. South Africa, Johannesburg [ˈneɪkʰəd]
36. Australia, Perth [ˈnɛɪkəd]
37. New Zealand, Auckland [ˈnɛɪkəd]
38. India, New Delhi [ˈne̝ˑk̟ɪd]
39. Singapore [ne̝ɪkəd]
40. Nigeria: [ˈne̞ˑkəd]

See how much variation there is around the world, even around Britain herself?