There is no authoritative answer to this question. The Palestinians, as far as I know, don't have clear census information even about the people currently living in the West Bank and Gaza, let alone the many hundreds of thousands of Palestinians in the rest of the Arab world, and in other countries outside the Middle East. On the contrary, it has always served Palestinian political leaders and their propaganda to exaggerate the numbers of Palestinians of all ages, so that by some estimates there are well over seven or eight million of them. Another problem is that according to standard Palestinian definitions, anyone no longer living in his or her original home, even if he lives somewhere else within the Palestinian territories, is a refugee: this is rather like saying that if someone originally from New York City has been forced to move to Boston, Rochester or New Haven, then he and his family for untold future generations are refugees. And since the Palestinians have consistently insisted that people in this situation, even if living only a few miles from their original homes, are refugees, and shouldn't simply develop a new permanent life wherever they are now, they have both milked international public sympathy and international donors for help.
However, by definition, any original Palestinian refugee must be over 58 years old, so it is a little easier to make an educated guess. The original Arab population of Palestine (west of the Jordan river) in 1948 was about 1.5 million, and probably one-third of them became refugees in Jordan and other countries. If we assume that those who were then younger than 30 (up to 88 today) made up half of the refugees, that makes something like 250,000 people.
I hope this answers your question.
David