I chuckled at this because I've had the same experience many times over. Because I deal with so many elderly people who are hard of hearing, and because many of them have had similar thick Southern accents, I know exactly what Lily Beth was doing. I even remarked about this phenomenon at a recent funeral I performed for a dear lady of our church.
When they can't understand you, mirror their accent and your speech will become clear to them.
This isn't only true for the hard of hearing. Years ago, when I lived in the Roanoke valley of Virginia, I found this to be true in sales. I didn't do it intentionally, but when I was trying to sell my product to a customer often I would adopt a similar accent to theirs. The result was that it put the customer at ease and established rapport. Lily Beth says that during the years that we lived in the valley, my accent changed. It became thicker than hers--and I presume that this was because I was doing sales and she was not. I had a reason to change my accent, and she did not.
Come to think of it, this principle is true when it comes to spiritual conversation as well--and not only in terms of adopting another person's speech patterns. In 1 Corinthians 9.20-22 (NIV), the apostle Paul says:
20 To the Jews I became like a Jew, to win the Jews. To those under the law I became like one under the law (though I myself am not under the law), so as to win those under the law. 21 To those not having the law I became like one not having the law (though I am not free from God’s law but am under Christ’s law), so as to win those not having the law. 22 To the weak I became weak, to win the weak. I have become all things to all people so that by all possible means I might save some.
In spiritual conversation, Paul employed this same principle that I discovered with people's accents. By establishing commonality with a person, you gain their trust. Trust breaks down barriers of understanding, and makes it more likely that people will receive your message.
Now, don't get me wrong. I'm not saying that you should become something you're not. Don't lie to a person in order to establish that commonality. Beth and I are both Southern, and we both have slight Southern accents. At various times, she and I have accentuated our Southern accents, either intentionally or unintentionally, in order to make ourselves more easily understood by Southerners. It would be quite another thing to travel to London and try to pass ourselves off as English! Paul was Jewish, so he could accentuate his Jewishness when he was dealing with fellow Jews. Yet, he also felt himself free in Christ, so when he was talking with those not under the Jewish law, he made himself like them. In neither of these cases was he pretending to be something that he was not. To the weak, Paul accentuated his weaknesses, but I imagine that he spoke with confidence to ruling-class people of high esteem. When I'm with fishermen, I talk about fishing (which I love). When I'm with hunters, I talk about hunting. Even though I've only been hunting a few times, it's enough--and it establishes rapport.
Do you have someone you'd like to share Jesus with, but you've found that your message has not been understood. Maybe you've been speaking perfectly clearly but like someone who's hard of hearing, they've been asking you to repeat yourself. Don't put on a show and try to be something that you're not. But if there's something that you are, and accentuating that can help you communicate more clearly with them, then by all means accentuate your accent. It might make the difference between them simply hearing and actually understanding.