The American Heritage Dictionary seems to see this as stemming (hehe) from when the Old English preposition on was placed in front of a verbal noun:
Prefixing a- to verb forms ending in -ing, as in a-hunting and a-fishing, was once fairly common in vernacular U.S. speech, particularly in the highland areas of the South and in the Southwest.
Such verb forms derive from an Old English construction in which a preposition, usually on, was placed in front of a verbal noun — a verb to which -ing had been added to indicate that the action was extended or ongoing. Gradually such prepositions were shortened to a- by the common linguistic process that shortens or drops unaccented syllables. The -ing forms came to be regarded as present participles rather than verbal nouns, and the use of a- was extended to genuine present participles as well as to verbal nouns. Eventually a- disappeared from many dialects, including Standard English in the United States and Great Britain, although it is still retained today in some isolated dialect areas, particularly among older speakers. Today, speakers who use the a- prefix do not use it with all -ing words, nor do they use it randomly. Rather, a- is only used with -ing words that function as part of a verb phrase, as in She was a-running.
If this is true, then the original sense of the usage would have been a preposition-noun construction. In the song
A-hunting we will go
A-hunting we will go
Heigh-ho the dairy-o
A-hunting we will go
this means that "a-hunting" would have been used in the sense of "at hunting," meaning "being engaged in the pursuit known as hunting." It would not, then, have stemmed from the ME y or Germanic (and OE) ge verb form.
Usage note. AHD goes on to use the isolano of Smith Island in the U.S. as an exemplar of this kind of speech:
Residents of Smith Island speak a variety of American English unlike any other, fostered in long-standing isolation from other varieties. This unique dialect has its roots in the British dialects of the latter half of the 1600s, when the first English speakers set up residence on this group of islands. It is characterized by the retention of a number of older language features, including the use of hit for it, the use of an a- prefix on -ing verbs, and the use of phrases such as of a night for at night.
This is an interesting question. I'm not sure it has a definitive answer, but I thought I would add this into the mix just to get it on the record.
@Vitaly found an article ("American Speech", Duke University Press) that references some corroborative material. Here is an excerpt:
If, as according to Bybee and Dahl (1989) and Mittendorf and Poppe (2000), the progressive in English arose, following a universal linguistic tendency, from locative prepositional phrases, it was the erosion of prepositional on before gerundives with the -ing sufﬁx in such phrases that led to a-preﬁxing on participles.
It's a long article and I haven't digested it all yet, but it's very interesting.
More info furnished by @Vitaly: From The Progressive in Modern English: A Corpus-Based Study of Grammaticalization and RelatedChanges:
In EModE, the nominal origin of the to be a-hunting construction apparently still had an impact on its linguistic contexts. In the EModE part of the Helsinki Corpus, all instances of this type are found in intransitive contexts (regardless of whether they are introduced by the preposition in or by a-) (Elsness 1994: 22). Elsness (1994: 22) has explained this as “evidence that the -ing-form in these 160 The Progressive in Modern English cases was felt to retain some of its gerundial status: even though gerundial verbs may also take objects, such objects are less likely in constructions of the original prepositional type”. Such a trend is, however, not apparent from the ARCHER data: Out of the ten instances, five are intransitive, while three occur with a direct object (as example 128), and the remaining two are used with a prepositional object (such as example 129).
(128) Here have I been a treating the case as one of a garden, and jest as I’m a goin’ to pass sentence on the malefactors, you tell me they [the supposedly stolen peas, S.K.] were growing in a field! (archer\1800- 49.bre\1845surt.f5)
(129) Soldiers of France - The eyes of Europe are a-looking at you! (archer\1850-99.bre\1867robe.d6)
Apparently, then, the reluctance to use the prepositional type with an object vanished. This development is probably connected to the fact that a- was no longer interpreted as a preposition but as a prefix. As a consequence, the following ing-form was identified as verbal rather than nominal, in analogy to the much more common progressive.
More simply put, it appears that although the prepositional construction was present at the beginning, it shifted over time due to the bias of the general ear toward hearing the -ing form as a progressive verb rather than as a noun (gerund).