Of more difficulty is the name of the son. Manuscript H gives us Amr, while Manuscript A, which is likely to be the oldest, gives us Anir, either of which we might assume to be a mis-transcription of the other, however, manuscript E gives us Amir, mixing the two spellings.
The only other mention of this son comes from the mediaeval Welsh "Mabinogion" tale Gereint in which the son is named as Amhar. This is similar to "Gamber" if we assume some mis-transcription or evolution, however, this reference seems to have led most translators/scribers to give the name as Amr, presumably under the assumption that the Historia manuscript variations are mis-transcriptions of the letter "m" (though on this criteria one might equally suppose the original name Anihar or Anibar).
Amr is to a certain extent backed up by the name "Gamber" and the fact that Blaenau Gwent County Borough Council note that the Welsh name for the Gamber is Afon Amr* (though other evidence for this river name remains elusive). Amr certainly appears as a name constituent, for example in the name of Arthurian hero Bedwyr's son Amren, mentioned at the same time as Amhar in Gereint but also mentioned in in the court list and as fighting the Black Hag in Culhwch ac Olwen (see here for more about this story). One might suppose the two characters to once have been one, but there is no evidence for it - they may simply have been tied together because of the sound of their names.In Gereint we find out Amr and Amren were two of the four squire-chamberlains who guarded Arthur's bed. Some have speculated that this indicates a lack of favour, and possibly that Amr was illegitimate. Given the complex systems of kinship and guardianship which were in place in Dark Age Wales it seems a speculation too far - particularly given that the position was also occupied by Arthur's friend's son. That said, the killing of Amr by Arthur noted in the Historia does have a linguistic resonance if the name is, in fact, Amhir, as this term tends to start Welsh words associated with disrespect, rash abuse, and woe.
Equally, however, it is possible the name of the spring came before the Arthurian myth was attached to it. Amr is part of some words associated with eyelids (Amrant) - though more generally Am tends to act like the English "un-" or "en-", as, for example, in "encircle": amgylchu (cf. cylchu "circle"). On a completely different tack, Stevenson notes that Sir S. Meyrick suggested the original was actually Llygyd Aur "Golden Eye"*.
One might also note the rock Main-Amber [Maen "Rock": "Rock of Amber"] referred to by Camden*(Cornwall). This rocking-rock was apparently destroyed by an over-zealous iconoclast in the Civil War* (though it may have been a natural feature it obviously attracted visitors). Hunt* (online) quotes C. S Gilbert's An Historical Survey of Cornwall (1817) in suggesting that the name relates to the Cornish for "anointed" or "consecrated", though evidence for this is not forthcoming.
In fact, Anir itself is actually a perfectly decent word in Welsh. Though its meaning of "old" or "(age?-)shortened"* doesn't particularly make sense in the context of the Arthurian story, there are springs in Wales named Hen - again meaning "old" (one, for example, was used as the boundary of the Dark Age kingdom of Gwynlliog in South Wales [see the Life of St.Cadog]). If the original name of the spring was "Old Spring", it would be very old indeed.