There is a website (here) that Identifies an ancestral McHenry family of Antrim, Ulster - specifically to 1590 and the birth of a Daniel McHenry in Kinbane Castle in Antrim County, Northern Ireland to a member of the O'Cahan Clan. This information would confirm the McHenry family in about the same place and about the same time as our Jameson family can be found in Ulster.

It may also be interesting and amusing that this same webpage speculates of this, above named Daniel McHenry as: "Further research shows that Sir John Perrot, the Lord Deputy of Ireland and illegitimate son of Henry VIII, was active in Northern Ireland during this time and was reputed to have taken a mistress from the local Irish population.  If so, it would explain why his bastard child, Daniel, was given the new surname McHenry, which is Irish for "Son of Henry."