Which of the following phrases is correct: "to come to mind" or "to come to one's mind"?
If both are possible, do they have the same meaning?
If not, when should I use each of them?
Would you please give me some examples?
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We do not typically say this phrase with "one's" in it. Just use "come to mind," in whatever tense suits the context. Really, you won't hear it any other way.
"Joe, can you think of any way to make this meatloaf taste good?"
"Hmm. Gosh, no. Nothing's coming to mind."
This is probably the most common way of using the phrase, as an equivalent to "think of something/anything." ("Hmm. Gosh, no. I can't think of anything.")
Modifying my answer to better reflect the question...
They mean the same thing, and is a generic statement meaning approximately "it occurs to me." That being said, neither form is commonly used; the typical way to express this in this form is "It comes to mind...", leaving off any possessive.
In response to @Epiphany, the use of "What comes to one's mind" is not to expand the meaning to known or unknown persons ("ones") who may or may not be present to have something occur to. For this to be the meaning would require mind-reading -- how do you know it occurs to someone else that such and such might be the case?
The "It comes to mind" expression has always been an alternative way to express the idea that some intuition or inspiration has come to a person -- others are not involved.