As Sunnyside is common as a house-name, to the point of being something of a cliché, it seems likely that this street was named after a house. For example, there was a house of this name on Windsor Avenue in the 1880s, although the location is too far from Sunnyside Street to account for this particular name.
There are references in newspapers in the early 1870s to a row of houses called Sunnyside at Ballynafeigh (an alternative spelling of the townland Ballynafoy). It is possible that this is the origin of the street name. Further information would be very welcome.