There appear to be between 6 and 8 women in the New Testament named Mary (please call out any I've missed!):
Mary the mother of Jesus, wife of Joseph, from Nazareth (e.g. Luke 1:26-27)
Mary Magdalene, (probably) the first witness of the resurrection, from Magdala (e.g. Luke 8:2-3)
Mary of Bethany, sister of Martha & Lazarus (e.g. John 11:1-2)
Mary the wife of Clopas (or Cleopas), a witness of the crucifixion (e.g. John 19:35)
Mary the mother of James & Joses, a witness of the crucifixion & resurrection (possibly the same person as #4) (e.g. Mark 15:40, Luke 24:10)
Mary the mother of John Mark (see Acts 12:12); her home in Jerusalem was apparently a meeting place for Christians. Her son traveled with Peter & Paul and repeated patristic evidence indicates that he wrote the second Gospel
Mary of Rome (see Romans 16:6); we know precious little about her, but she was apparently a source of help to Paul, and she appears to have traveled (Paul had never been to Rome--at least not as a Christian preacher--when he wrote this)
Mary the sister of Mary? This one might be a stretch; it depends on the punctuation of John 19:35:
A. Now there stood by the cross of Jesus his mother (1), and his mother’s sister (2), Mary the wife of Cleophas (3), and Mary Magdalene (4) OR
B. Now there stood by the cross of Jesus his mother (1), and his mother’s sister Mary (2), the wife of Cleophas (3), and Mary Magdalene (4)
By comparison to Matt 27:56 some have concluded that the wife of Cleophas is the mother of James & Joses, and thus have concluded there were 4 women named Mary at the cross. B seems odd because it would mean a family named 2 daughters Mary...stranger things have happened though.
See a source for much of this information and more detailed discussion of seven Mary's here.
The time and place
Richard Bauckham (here), Peter Williams (here), and others, have pointed out that from a tally of surviving sources, including written documents & inscriptions, Mary was the most popular Jewish girl's name in the first century in the region where Jesus lived. It is the same name as Miriam in the Old Testament.
An interesting point explored by both Bauckham & Williams is what this means for the writing of the Gospels. I've shared thoughts elsewhere on the site regarding the means by which the Gospels are dated (see here); this one is interesting specifically with respect to eyewitness sources.
If a Gospel author had been making up the stories, it is extraordinarily unlikely they would have named so many characters Mary (or Simon for that matter). The prevalence of the right names speaks to the stories' early date and reliance on eyewitnesses.
Why are there so many Mary's in the stories of the New Testament? Because there really were so many Mary's.