The Catholic Church considers any marriage involving at least one Catholic party to be subject to Canon Law. The primary objective is to safeguard the Catholic spouse’s faith. A marriage between a Catholic and a non-baptized person is considered to be invalid under Canon 1086.1, generally speaking. However, since this canon is ecclesiastical law, not divine law, the appropriate ecclesiastical authority has the power to dispense from this law. Therefore, this impediment - the disparity of cult - can be dispensed by the local ordinary (diocesan bishop) if he decides it is justified in a particular case. Once the dispensation is obtained, a Catholic can validly marry a non-baptized person.
(The law is different regarding the marriage between Catholics and validly baptized non-Catholic Christians, but I will not get into that here as it is not relevant to your case.)
As I said earlier, the Church makes is difficult for Catholics to marry non-Catholics because the primary concern here is for the faith of the Catholic spouse. And with this in mind, it is only logical for certain requirements to be met before the bishop can dispense for disparity of cult:
The Catholic spouse must promise to do everything in his / her power to ensure that the children are raised Catholic.
The non-Catholic party must not be hostile to Catholicism and must allow the Catholic spouse to continue to practice his / her faith after the marriage. The non-Catholic must also not be hostile toward the idea of raising his / her children Catholic.
Most of the time people who obtain this dispensation from the bishop do this through the pastor at the Catholic spouse’s parish. First, the pastor must decide that it is appropriate to ask for this dispensation, and usually the bishop will dispense unless he disagree with the pastor’s assessment of the situation.
Civil marriages are not allowed except when permission is granted (or another dispensation obtained, I don’t remember which). I’m not entirely sure of all the details involved here, but a dispensation for disparity of cult is not an automatic “permission” of marrying in a civil marriage. That would be a defect in canonical form.
I’m not sure what you mean by getting a marriage “blessed.” The idea of getting a marriage blessed does not exist in Canon Law, and most people are indirectly referring to radical sanation / convalidation. It doesn’t appear to be relevant here.
In short, your son can only marry an unbaptized person if he obtains the dispensation for disparity of cult, most likely from the diocesan bishop through his pastor.