A temporary marriage contract cannot be "converted" to a permanent marriage. What would need to be done is the man says to the woman he is in a mutah contract with that he frees her for the remaining period of their contract, and they can then do a permanent marriage contract.
Of course, they would need to agree on a mahr as well, and there is no 'iddah period, because he will be marrying the same person he was with. This means that once he frees her of the remaining mutah period, they can straight away do the permanent marriage.
This is unacceptable. Whatever the family's religion may be, Muslims must honour and respect their families. Your wife has no right to do this.
If she has certain concerns, you should address them. There might be something she has seen, like abuse, or intimidation, or trying to brainwash your son, etc... Try to pinpoint what the dispute is about, and deal with it.
But if it is just because she has no respect for your family, this does not give her the right to deprive her son of visiting or being with his grandparents and family.
If all other shar'i requirements are met, yes, she can marry that man.
The marriage proposal must take its normal procedure, primarily around the parents giving their consent and their blessings. This is the most important part.
A second important step is to see the character traits of the person, their moral standards, their family, compatibility, etc.
A third step to take is seeking consultation and advice, trying to find out if this prospect marriage is a good idea, and the obstacles they might face. This is necessary for any case of marriage, Muslim born, or convert.
Both of them should study this from all its angles, for the sake of making the right decision and having a successful marriage.
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