It can be spelled both ways is the short answer. Crape/crepe refers to the crinkled edges of the flower, crape is the older of the two ways of spelling but I am not sure that this makes it correct. The American Crape Myrtle Society spell it with the 'a' so I would use that but I think I would spell it with an 'e' if writing for Southern living though.

I am not going to do the research Ink has done but generally it is spelled "crape" myrtle but it seems to be one of the most commonly misspelled words in gardening. I think we need a forum or at least a topic on most commonly misspelled garden words. Down here in Dixie (can I still use that term?) Most people when writing Callaway Gardens spell it Calloway Gardens. They are simply not paying attention in my opinion. That, or correct spelling is simply not important to them.
Not being a good typist I spell more words wrong that way than when I am writing longhand.

I enjoy spelling it "crape" because to me that's the more traditional English spelling. True, it's a word of French origin, but has been thoroughly naturalized in English with the spelling crape, especially for the cloth -- and I believe it was the likeness in texture to the cloth that gave rise to the name crape myrtle.

There are numerous other such words of French origin, now thoroughly naturalized in English. Another example is envelope, which in fact is no longer a French noun -- the French is enveloppe -- and which sounds less pretentious with the first syllable pronounced en-, not on-. I am noticing more and more half-baked French pronunciations like this lately, especially among younger people, often those who have little knowledge of real French pronunciation.

I guess that's rather irrelevant, but I liked getting it off my chest!

The French for crape, by the way, is crêpe. But I have to admit that the spellings crape and crepe both found their way into English.
[sorry, I tried to spell crepe with an e-circonflexe, but it refuses to show in the message preview -- likewise with the acute and grave example at end]

The general rule for spelling and pronunciation of these French-origin words, as recommended in guides such as the New Fowler's Modern English Usage, is that thoroughly assimilated words should be treated as English, while words still in the process of assimilation can be treated as French. That reference draws attention to the word "abandon", which did not appear in English until 1822 and was still given a French pronunciation in the early 20th century. And if you insist on treating words as French then it behoves you to add all the right accents, e.g. "ménage à trois".

The oldest text book I have, spells the word both ways in reference to Lagerstroemia. I learnt it as crepe like in 'crepe paper' (always spelt this way?) because the flowers looked as though they went through a similar process. Additionally, 'crape' has meanings associated with crepe and curly. But I have been called a poor speller because of spellings like crepe myrtle, programme, sulphur etc.

You overlooked colour. I have always thought of crepe as the paper and crape as the plant.

It's always been crepe myrtle here until recently.

crepe de chine
crepe paper

I like the look of crepe better than crape, which always looks to me like a 4-letter word with an e at the end.

The traditional way is crape, like it or not. It's a common name, not scientific, but it's worth noting that with scientific names, the original name, even if incorrect in some way, is correct. Thus spake Linnaeus.

For what it's worth, something I wrote for the San Antonio Express-News in 1995

We say crape; you say crape

Every time we write about crape myrtles, the clucking and finger wagging starts.

Don't we know it's crepe myrtle?

Then the editors thump their official dictionaries Webster's New World Dictionary, Third College Edition where it's crape myrtle.

So why would knowledgeable nurseries and respected authors and botanists spell it with an e instead of an a.

And why would the official dictionary's definition say "so named because of its crinkled, crepe like flowers"?

" Crape myrtle is the overwhelming choice both in botanical sources and in other dictionary sources," said Michael Agnes, executive editor of Webster's New World Dictionary.

The word expert explains that crape myrtle is a compound, two elements referring to one thing. "When that happens, a variant spelling is almost always associated with the compound," Agnes said.

But which vowel came first?

The first reference to crape, without the myrtle, came in 1685. Crepe first showed up in 1797, Agnes said.

"By the time someone decided to call this plant crape mytrle, crape was by far preferred," he said. The first reference to crape myrtle showed up in the Oxford English Dictionary in 1850.

"It's just one of those wonderful word wonders," Agnes said.

And the good thing about it neither crepe nor crape is wrong.

San Antonio Express-News

For many decades I saw it spelled with an "e," but in recent years, I see it spelled with an "a" more frequently. The language of writers is changing. English is to be spoken and written to accommodate California (even in our textbooks), making it easier for many different languages spoken there, or at least that is what the academics are telling those of us who are still interested in language.

I just wish broadcast journalists and writers would remember what they were taught in grammar school regards subjective and objective pronouns.

Since "Crape" is the oldest, more traditional, and more authoritative, I would go with that one. It has the meaning of something being crinkled. "Crepe" should have never snuck in for the meaning for the tree or crinkled cloth. Even if we have all spelled it wrong over all these years, because of our familiarity with the Original Pancake House, we will just have to relearn that crape and crepe are two different words. I do love the beauty of the Crape Myrtle at the end of summer, and I love the French Crepes at the Pancake House. Here is an explanation: https://brians.wsu.edu/2016/05/31/crape-crepe/