It has nothing to do with Cleopatra. The needle which is one of a pair (the other one now being in New York), was commissioned by Pharaoh Tuthmosis III, 1,400 years before Cleopatra was born.
Cut from red granite, the needles were originally erected, uninscribed, outside a temple in Heliopolis. Two hundred years later Ramesses II had inscriptions carved of them, something he did to any spare stonework that didn’t already make mention of his great deeds. It was only under Roman rule, in 12 BC, 18 years after Cleopatra’s death, that the obelisks were taken to Alexandria, the city that had been her capital. Here they were erected outside the Sebasteum, a temple which Cleopatra had begun building in honour of Mark Antony.
In 1801 the Ottoman viceroy of Egypt gave Britain a needle in thanks for victories over the French but it was only in 1877 that an English dermatologist paid for it to be delivered. It arrived in 1878 after nearly sinking in the bay of Biscay and was erected on the Embankment in London.
So how did it get the name? That was probably borrowed from the Cleopatra’s Needle already in the Place de la Concorde in Paris. That had originally come from Luxor and also had absolutely nothing to do with Cleopatra.