The earliest attested spelling of the word bald seems to be balled, as you have noticed. According to the Oxford English Dictionary (OED), "Middle English balled" is "of uncertain origin".

However, the use of "bald" to describe animals that have white streaks/spots/markings is attested well before the specific term "bald eagle". So it seems quite plausible that the bald eagle was simply named this because it was bald, as in, marked with white on its head. However, it is probably also not a coincidence that this white head resembles a hairless (bald) human head.

The OED gives it as the fifth definition:

with the following examples:

The American Heritage Dictionary (AHD) gives it as the fourth sense:

Zoology Having white feathers or markings on the head, as in some birds or mammals

So I would say, just tell your student that it is a specialized/obsolete use of "bald" that most speakers are not familar with.

This sense of bald also seems to be the origin of the word piebald (which occurs in one of the OED examples above). The AHD entry for piebald says:

PIE2 [as in "magpie") + Early Modern English bald, marked or streaked with white.

Thus, bald eagle and the word piebald seem to be indirectly related. However, neither the OED nor the AHD seems to be in favor of describing "bald eagle" as having been directly derived from "piebald" (an etymology that Michael Seifert mentioned in a comment); i.e., I can't find any evidence that people used to call this animal a name like "piebald eagle" and then that was later shortened to "bald eagle".

The origin of bald in this sense seems especially unclear. It might be derived from the noun ball, which apparently doesn't have an attested cognate/ancestor in Old English but which seems to have cognates in some other Germanic languages. The OED says "ball" is

Probably the reflex of an unattested Old English *beall (compare Old English bealluc bollock n.), cognate with Middle Dutch bal ball, sphere (Dutch bal ball, sphere, ball of the foot, heel of the hand), Middle Low German bal ball, ball of the foot, heel of the hand, Old High German bal ball for playing, small sphere, mouthful (Middle High German bal ball, globe, ball of the foot, German Ball), Old Icelandic bǫllr ball, sphere, hill, Norwegian ball ball, Old Swedish balder, baller ball (Swedish boll, (regional) ball, balle), Old Danish bold ball (Danish bold ball), and further with Old High German ballo, pallo (weak masculine) ball of the foot, heel of the hand, mouthful, ball for playing (Middle High German balle ball, sphere, ball of the foot, heel of the hand), and Old High German balla, palla (weak feminine) ball for playing, mouthful < the same Indo-European base as classical Latin follis inflated ball, bellows, and (with different ablaut grade) ancient Greek ϕαλλός penis, image of the (usually erect) penis, especially as a symbol of the generative power in nature (see phallos n.). Compare bale n.3 for forms in Romance languages probably borrowed ultimately < the same Germanic base.

However, as you note, several sources suggest bald in the particular meaning "having white spots/streaks/a white spot/a white streak" is instead related to, or in some way also associated with Welsh "bal":

OED:

Compare Welsh ceffyl bàl a horse with a white streak or mark on the face (French cheval belle-face), where bàl may be an adjective, or a noun construed as a genitive.

AHD:

Sense 4 ["Zoology Having white feathers or markings on the head, as in some birds or mammals"] perhaps partly of Celtic origin; akin to Welsh bal, having a white streak on the forehead (of horses), Irish ball, spot, mark, and English blaze, white mark on the face of an animal; see BLAZE2.

The word blaze is traced back by the AHD to the PIE root

‌‌ bhel-1 To shine, flash, burn; shining white and various bright colors.

Which is identical in form with, but treated as a distinct root from

‌‌ bhel-2 To blow, swell; with derivatives referring to various round objects and to the notion of tumescent masculinity.