Checking the records, it seems that Arthur Street doesn't appear on the 1685 map or the 1737 one, but is there in 1833. (On the map it runs roughly north-south, Upper Arthur Street being the southern end which runs off the bottom edge of the map, just east of the church marked A67 [the Donegall Square East Methodist Church].) The river running through the area in the 1685 map is the Blackstaff, before it was diverted to enter the Lagan a bit further upstream, allowing houses to be built in Arthur Street / Victoria Square.
Well, Google Books gives me partial access to Marcus Patton's 1993 booklet, Central Belfast: An Historical Gazetteer, enabling me to piece together the following sentence:
Eliza Street was probably named after Lady Elizabeth, daughter of the First Marquis [sic] of Donegall, with Charlotte Street, Amelia Street and Arthur Street being named after her siblings.Debrett's, available in full from Google Books, tells me that the first Marquess of Donegall (whose own dates were 1739-1799) had seven children by his first wife (and none by his second or third), listed as George Augustus (1769-1844), Arthur (1771-1788), Spencer Stanley (1775-1819), and the girls whose birthdates are discreetly omitted, Charlotte, Anne Henrietta, Elizabeth Amelia, and Emilia, all of whom apparently died young.
I'm not sure that I buy this story. Eliza Street, Charlotte Street, Amelia Street, and Henrietta Street which would also go with this pattern, are all to the south - in the case of Charlotte Street, some way to the south - of the Howard Street - Donegall Square - May Street axis where Upper Arthur Street ends. It does make sense that the marquess, who basically owned Belfast, would commemorate his dead daughters in the streets built in the later eighteenth century; but looking at the geography, Arthur Street should have been constructed a few decades earlier. I think it's more likely that it commemorates the first marquess himself, rather than his son, as his name was also Arthur (as were all four of his predecessors as Earl of Donegall).
In any case, it's pretty clear that there is no connection, or only a distant one, with the Presidential family of the Ballymena area. I remain on the lookout for local connections to the 21st President; next time I'm up in Derry, I shall ask my old friend Paul if he has any ideas.