If the generic name changes for these birds and they become Buteo, what happens to taxa which have the same endings such as Leucopternis albicollis costaricensis and Buteo jamaicensis costaricensis ? I suppose the former should change as it is the youngest.
Yes, this is called "secondary homonymy" - the older homonym has priority, and another name is needed to replace the younger one.
(But note that the homonymy persists only as long as the two names are kept in the same genus. If you take one of the species back out of this genus subsequently, you must revert to the older name for the taxon that had its name replaced. Failing to do so is a relatively frequent mistake.)
If you extend
Buteo to such an extent that it includes
Leucopternis, you get a triple secondary homonymy, actually:
-
Buteo borealis var.
costaricensis Ridgway, 1874
-
Leucopternis ghiesbreghti costaricensis W.L. Sclater, 1919
-
Asturina nitida costaricensis Swann, 1922
Ridgway's name applies to the race of the Red-tailed Hawk and has priority.
The Gray Hawk has for a long time been placed in
Asturina by some authors, in
Buteo by others, and a replacement name has been proposed for Swann's name:
Buteo nitidus blakei Hellmayr & Conover 1949.
But
Leucopternis is a quite different case: this genus has
never been merged with
Buteo since the proposal of
costaricensis by Sclater. (In fact, the definition of buteonine generic limits has traditionally rested on a
Buteo vs.
Leucopternis dichotomy; e.g., the usual reason cited in support of the recognition of
Asturina has been that it was possibly closer to
Leucopternis.) I doubt another available name exists.
Where could I find the next available name for it then?
You can try the WorldBirdInfo website, as this is a group that is covered there - go to the species' page, click "Display Citations, Note & Scientific Synonyms", and scan the citations for a synonym that could be used.
Otherwise, for American birds, Cory & Hellmayr's
Catalogue of the Birds of the Americas and the Adjacent Islands usually has good synonymies, and is conveniently available online (e.g.:
http://www.biodiversitylibrary.org/Search.aspx?searchTerm=birds of the americas&searchCat=).
But neither cites a name that could be used in the present case.