Offhand, the only times I recall hearing this phrase is when folks defend homosexuality. How should Christians respond?
i) Open theists think God does make mistakes. At least, that's what their theology commits them to. A God who doesn't know the future will be mistaken about the future. He's bound to get some things wrong.
ii) You also have deistic evolutionists who believe God created the initial conditions, but after that, the evolutionary process takes on a life of its own. God doesn't guide it. He may still intervene in human history, but not in natural history.
Technically, he doesn't make mistakes. Rather, he creates a trial-and-error process.
This is also true for theistic evolutionists in the sense that random mutation is an essential component to evolution, and mutation involves transcriptional errors. Technically, God doesn't make mistakes. Rather, he creates a process that's liable to replication errors. And that's not incidental to the process. That's essential to the development of life by evolution.
So even if you deny that God makes mistakes, assuming you you subscribe to evolution, then homosexual orientation could well be a mistake. Indeed, anything as statistically abnormal and disadvantageous as homosexuality is an excellent candidate for one of nature's many dead ends.
iii) Likewise, there's such a thing as genetic defects. Homosexual apologists can't wish that away by saying God doesn't make mistakes.
If, say, a patient is diagnosed with a surgically correctable congenital heart defect, are they going to deny the condition or refuse the operation on the grounds that God doesn't make mistakes?
iv) Keep in mind, too, that since homosexual apologists typically claim that homosexual orientation has a physiological basis, once they allow for random mutations and birth defects, they can't very well exclude the possibility that homosexual orientation is a "mistake."
v) Finally, let's examine the concept of a "mistake." Take planned obsolescence. It's possible to make some products more durable. But that's less profitable than built-in obsolescence. It's a designed defect rather than a design defect. The fact that it isn't as good as it could be is intentional.
Let's take another example. Take a tactical defeat (or tactical loss). An up-and-coming poker player may deliberately overplay his hand a few times in a game to make his opponents underestimate him. He intentionally loses a few hands to make them overconfident. They think he's easy to beat. That's their downfall.
Or this can operate in reverse. In the The Cincinnati Kid, Lancey plays an aging poker player who used to be the best in the business, but pretends to be losing his touch in his match with Stoner. Stoner is lured into thinking Lancey is past his prime. That causes him to tempt the odds, which plays right into Lancey's trap.
Lancey's "mistakes" are calculated to embolden Stoner. That gives Stoner the opening he's been waiting for. Except it's really an ambush. Lacy lets Stoner to win just often enough to lose when he's got everything on the line.
The fact that we live in a fallen world inhabited by a variety of sinners is not a mistake on God's part. That was his plan all along. It's a means to an end.
Indeed, the position of the homosexual apologist is self-contradictory. If God makes no mistakes, then they can't say the Bible is homophobic. For that imputes error to the word of God.