Does this mean the correct pronunciation is "deekstra"?
Daikstra - you can pronounce this
Dijkstra - a sound in between you cannot pronounce coming from English
Diekstra - you can pronounce this too
But the 'ɛi' is a dipthong, and we don't have that dipthong. We have aɪ, as in 'bride' /bɹaɪd/, which is what usually comes out, at least for those of us who know that /dʒɪkstɹə/ is not even close to right. What you'll hear is generally something like /daɪkstɹə/, which make sense, as we pronounce 'dike' /daɪk/ as well.
So yes, completely correct, I just wanted to add the IPA for clarity.
[0] The exceptions being dialects where it monophthongizes to /eː/
Phonemes are a funny thing: if an American were to pronounce that last name perfectly (or a Dutch person speaking English, for that matter), other Americans would still hear /daɪkstɹə/, or at most, "/daɪkstɹə/ but it sounds a bit foreign".
Some sounds which we don't produce are audibly not-English-phonemes, like German ü or Scandanavian ø, but others are not. Another example of this is that native English speakers aren't usually sensitive to the difference between the r sound in our word "pear" and the r sound in Spanish "pero", but recognize that "perro" is a different rhotic sound.
If I may ask purely out of curiosity, are you a native English speaker? I only say this because, to my American English ear, [eɪ] and [ɛi] are virtually indistinguishable. Maybe I'm just misunderstanding what you're trying to say....
I encourage you to listen to a native Dutch speaker pronounce Mynheer Dijkstra's name, which you can do on Wikipedia: https://en.wikipedia.org/wiki/Edsger_W._Dijkstra
Does that sound more like "dike" or more like "day"? To me (yes, native American English speaker, of one of the dialects very close to 'Standard'), it is obviously more like the former.
I might've figured out the disconnect. According to this image on Wikipedia[0], the Northern Standard Dutch realization of [ɛi] starts lower than the monophthongal [ɛ] in the same dialect[1]. Just speculation on my part, but maybe in the context of Dutch linguistics, [ɛi] conventionally signifies a wider diphthong than I would've naïvely thought? (If so, I guess this makes me a cautionary tale against directly comparing IPA transcriptions between languages....)
[0] https://en.wikipedia.org/wiki/Dutch_phonology#/media/File:Du...
[1] https://en.wikipedia.org/wiki/Dutch_phonology#/media/File:Du...
Dai = Day
Die = Die
If you make an "ay" sound, then transition to an "i" sound, you can feel that the sounds are continuous. So maybe that inbetween sound OP is talking about literally the sound you make with your tongue between "ay" and "i".