Your friend’s mother has the responsibility to substantiate her claim by citing the Bible verse she says prohibits naming sons after their fathers.
This is mainly a cultural question and not a religious question. You are free to name your son whatever you and your wife agree upon as appropriate.
As a side note though, please be aware that titles like John IV and John V are supposed to be royal titles. Commoners drop the distinguishing numbers after the deaths of the forebears who were ahead of them in the family line. You are, of course, free to call yourself and your son whatever you wish; but if you are interested in the rules of usage for such matters, here they are:
John Smith’s son is John Smith Jr. His grandson is John Smith 3rd (Roman numerals are a royal affectation). When John Smith dies, his son becomes John Smith and grandson John Smith Jr. The only person to use the title Senior is the first John Smith’s widow, in order to distinguish her from her daughter-in-law who is the new Mrs. John Smith.
In your family, unless your great-grandfather, grandfather, and father are all still alive, you would not be Daniel 4th, and your son would not be Daniel 5th. Your place in line would be determined by whichever of your forebears is still alive. For example, if your father is the only one of them still alive, he would be Daniel, you would be Daniel Jr., and your son Daniel 3rd.
