Demetrius is the Latin form of the Greek name Demetrios. And the name Demetrios is believed to be from the Greek goddess Demeter. In Greek, De means "earth", while meter means "mother". So, Demeter was the "earth mother". In greek mythology, Demeter was a rather important goddesses, she was the goddess of agriculture. The people were dependent on her to help them grow their crops.
Demeter had a daughter named Persephone, who was kidnapped by Hades. Demeter grieved and did not let anything grow. Hades did everything he could to please Persephone in the Underworld, and tried to make her eat. One day, in the gardens he allowed her to plant in, she was extremely hungry, and her garening helper boy offered her some pomegranate. Sadly, she ate six seeds, thus, she was only allowed to return to her mother every year for six months. The other six months she stayed with Hades. This is the reason why there are the different seasons, spring and summer when Demeter is with her daughter. Then, autumn and winter when Persephone is away.
Shakespeare could have had many reasons for using the name Demetrius. One possible reason is that both Demeter and Demetrius are denied the things they want the most. Demetrius longs for Hermia's love but does not get it, and Demeter wishes to have her daughter Persephone with her. Another possible connection to Greek mythology is that Demetrius is like Persephone. She was forced to stay in the Underworld for six months because she ate six seeds. Just how Demetrius was bewitched by the flower's sap to love Helena.