A425 : by Tony Garland 
Thanks for your ministry and your question.
I don’t think the answer to your question is found through appeal to the underlying Greek. Instead, it would appear the negative, “will not blot out,” is used simply because of the way in which the Book of Life is used.
The names of all the living—with the exception of the terminal generation of “earth dwellers” of the time of the end—are initially written in the book. This underscores the offer of salvation to all who would come to faith in the shed blood of Jesus on their behalf: “God so loved the world that He gave His only-begotten Son…” (John 3:16). While still living, unbelievers have the opportunity to respond in faith to the gospel and be saved.
Upon death, those who do not respond to God’s gracious offer of salvation are then blotted out of the book. As the opportunity to be saved comes to an end (Heb. 9:27) their names are expunged from the book. This is the default scenario for those who do not receive Jesus—the majority of people throughout history, as Jesus explained (Matthew 7:13-14). These, then, stand at the Great White Throne judgment where the Book of Life is used to verify they are not covered by the blood of Christ (lack the righteousness of [provided by] God, Rev. 20:15).
Unlike those on the wide path, believers respond and choose the narrow path. Unlike the majority, they are NOT blotted out. So being blotted out is the majority case and the negation of that norm characterizes those who exercise faith in Jesus. Hence they are not blotted out. In fact, the Greek is emphatic at this point, employing the strongest “double negative,” for “not” (οὐ μὴ [ou mē]).
You said, “A friend thinks that in Revelation 3:5 the Lord is threatening some people who are not overcomers with having their names blotted out of the Book of Life.” I would agree with your friend: that is precisely what the verse says.
The more relevant question is not whether non-overcomers will have their names blotted out of the book (they most certainly will), but who are these non-overcomers? Are they believers who don’t measure up . . . under-performers? Or are they unbelievers who have not overcome through faith in Christ (1 John 4:4-5).
It is my opinion—and I believe Scripture is plain—these are non-believers. The distinction between overcomers and non-overcomers does not identify two categories of believers: Christian achievers (overcomers) vs. Christian under-achievers (“undercomers” if you will). Rather, Jesus is writing to both believers and unbelievers within each of the seven churches of Revelation. Only believers are overcomers, and all believers are overcomers through identity with Christ. It isn’t about performance, it is about relationship.
Those who are “in Christ” remain secure and obtain the promises given to the overcomer. By the same degree, they avoid the implied negatives—the flip-side of the promises to the overcomer—coming to those who fail to overcome. The telltale clue, beyond John’s clear definition of overcoming mentioned above: overcomers receive promises which are equivalent with salvation (e.g., eating of the tree of life, avoiding the second death—the lake of fire).
An additional point, if I may. I would strongly discourage delving into the original languages behind the text by attempting to use Artificial Intelligence (AI). Here’s why: it has been shown that AI can be unreliable at times—even completely wrong. Those lacking training in the area of expertise being leveraged using AI are not equipped to properly evaluate when AI is going astray. This is as true for biblical languages as it is for other complex disciplines such as, for example, software development, practicing medicine, or practicing law.
There is a reason we expect pastors to have numerous years of training in Hebrew and Greek before we trust they have adequate knowledge to use the biblical languages safely. Much like doctors, lawyers, or engineers are licensed. And why? Because a little knowledge of a powerful tool is dangerous: one doesn’t know what one doesn’t know so-to-speak. It is akin to giving an automatic weapon to a teen without them having been trained in gun safety or having spent adequate time at the firing range. Destruction and damage is likely to result when the “tool” is accidentally misused. Alas, this is fairly common when untrained believers, however well-meaning, attempt to delve into the Greek or Hebrew with the belief they will gain marvelous insights not available in the English text. Their use of the original language is often problematic and mistaken—they don’t know what they don’t know.
This is one example where appealing to the Greek provides little, if any, help. Instead, an understanding of the wider context—considering how the book of life is used—provides more insight.
In closing, here are some resources that may prove helpful on the book of Life and the identity of the overcomer:
Blessings as you seek to understand God’s Word and continue to minister to others.
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