Since the remnants of Ernesto in the Atlantic Basin are likely to form a Tropical Depression (which will probably become Tropical Storm Hector) in the Eastern Pacific Basin, the following rules for storm names apply (from NHC):

Subject: B5) What happens to the name of a tropical cyclone if it moves from the Atlantic regions to the Northeast Pacific, or vice versa? Contributed by Chris Landsea

"The rule used to be that if the tropical storm or hurricane moved into a different basin then it was renamed to whatever name was next on the list for the area. The last time that this occurred was in July 1996 when Atlantic basin Tropical Storm Cesar moved across Central America and was renamed Northeast Pacific basin Tropical Storm Douglas. The last time that a Northeast Pacific system moved into the Atlantic basin was in June 1989 when Cosme became Allison.

However, these rules have now changed at the National Hurricane Center and if the system remains a tropical cyclone as it moves across Central America, then it will keep the original name. Only if the tropical cyclone dissipates with just a tropical disturbance remaining, will the hurricane warning center give the system a new name assuming it becomes a tropical cyclone once again in its new basin."

Since Tropical Depression Ernesto was classified as having dissipated over Mexico, the new system forming in the EASTPAC would get a new name. Ever since the new rule was implemented years ago, a tropical cyclone has never carried the same name from the Atlantic to the Pacific basin (or from the Pacific to the Atlantic basin).
ED