The name "River Gamber" is currently only used for the tributary of the Garren which extends to Gamber Head near Wormelow Tump - the furthest extent of the Garren watershed. The river issues here from Gamber Head Spring (Details - though caveat lector). From here the river travels south and is joined after a short distance at Trehumfrey by the Garren, which flows from around Garway to the north-west - a possible location for our wonder. At the head of the Garren proper is the suspiciously "Gamber"-like "Camdore" Farm, which may suggest the river name also once stretched up this tributary (the movement of the "G" to a "C" in Gamber is perfectly possible - for example, Kelly's 1856 Directory includes "Cummberhead"* for "Gamber Head"), however, it is more likely there is a relationship with the River Dore which runs in the next valley west into the Monnow, albeit that this is some 5km west of the farm, and over a hill).
South of the confluence of the two brooks, the river retains the name Garren (or sometimes "Garron") though Llangar(r)o(/e/a)n to Goodrich, where the river enters the Wye.
Blaenau Gwent County Borough Council note that the Welsh name for the Gamber is Afon Amr*, though other evidence for the use of this name remains elusive and it is unclear whether the usage extends to the Garren as well. While the Liber Llan Dâv* references the Am[h]yr and Evans* links this to the Gamber, it is clear in the indexing that he confuses potential references to the Gamber and actual references to the Humir near Newport (Note).