I'm wondering if the claim you heard was a conflation of a few things.
I could find two sources online stating that the title of Why Didn't They Ask Evans was something Christie overheard. The first is this IMDB review from 2013, which claims:
The title of the book actually came from a conversation Ms. Christie
overheard coming out of a movie theater, and she built a whole story
around it.
The second is this blog post:
In her autobiography, Agatha Christie talks about how she comes by
ideas for various books. This one started with the title, which was a
phrase she overheard.
I've read the autobiography, and have it on Kindle. I searched every variation of "ask", "asked", "question", "overheard", "Evans", "theatre", "title" and etc that I could think of and didn't find any reference to this particular novel or inspirations for it.
There is, however, a reference in the autobiography to one piece of inspiration Christie did get from overhearing a conversation:
I considered writing another book. Supposing I did--what should it be
about?
The question was solved for me one day when I was having tea in an
ABC. Two people were talking at a table nearby, discussing somebody
called Jane Fish. It struck me as a most entertaining name. I went
away with the name in my mind. Jane Fish. That, I thought, would make
a good beginning to a story--a name overheard at a tea shop--an
unusual name, so that whoever heard it remembered it. A name like Jane
Fish--or perhaps Jane Finn would be even better.
This became The Secret Adversary (1922):
“Funny scraps one does overhear,” murmured Tommy. “I passed two
Johnnies in the street to-day talking about some one called Jane Finn.
Did you ever hear such a name?”
As for the movie theater, take a look at this quote from Christie's Lord Edgware Dies (published a year before Why Didn't They Ask Evans):
We were passing a big cinema. People were streaming out of it
discussing their own affairs, their servants, their friends of the
opposite sex, and just occasionally, the picture they had just seen.
With a group of them we crossed the Euston Road.
‘I loved it,’ a girl was sighing. ‘I think Bryan Martin’s just
wonderful. I never miss any picture he’s in. The way he rode down the
cliff and got there in time with the papers.’
Her escort was less enthusiastic.
‘Idiotic story. If they’d just had the sense to ask Ellis right away.
Which anyone worth sense would have done—’
This is clearly someone overhearing a phrase, where the speaker is wondering why someone whose name begins with E wasn't asked--while coming out of a movie theater. However, the difference is that the overhear-ers here are Poirot and Hastings, not Christie herself.
It's certainly possible that Christie reused the phrasing from Lord Edgware Dies in Why Didn't They Ask Evans. However, I can find no evidence that she herself ever overheard this specific phrase.
