So, etymologically speaking, the word Elf or Aelfe is most likely derived from the proto Indo-European root word *albho-, which means white, through the First Germanic Sound Shift. This is evidenced in several languages' usage of cognates to *albho- and alβiz denote things that are white.

Close relatives are Latin albus (‘(matt) white’); Old Irish ailbhín (‘flock’); Albanian elb (‘barley’); and Germanic words for ‘swan’ such as Old English ylfetu. ^1

England was even named "Albion," literally the place of whites, in the Massaliote Periplus.^2

The jumpr from Alb to Alf to Elf is pretty understandable once you know that the Sound Shift actually continues to apply as words evolve; essentially, we get lazy with our articulation and slur letters into softer sounds.

1. Hall, Alaric. Elves in Anglo-Saxon England.
2. nēsos Iernōn kai Albiōnōn, the islands of the Iernians and the Albiones

TL;DR: You're assuming too much. Sources may be confusing Ælf- with other similar elements. More importantly, meaning is not so important in choosing names, even in Anglo-Saxon names.

Terms for anyone who isn't familiar with old Germanic-style names: these names are bithematic, being a protheme ("choose this from column A", so to speak) and a deuterotheme ("and choose this from column B"). A few themes could be found both as pro- and deutero-, but most are either one or the other. There was a limited set, though a few new themes were introduced and a few fell out of use. Bithematic names tended to be more common for higher classes and later in time; there were monothematic names, for example.

Searle writes, p. xiv, "At times the themes get confounded by the scribes, one being substituted for another. Thus Ælf- and Æthel- ... are found interchanged." P. 6 adds, "Ælf-, sometimes confounded with Wulf-". William Searle, Onomasticon anglo-saxonicum, is old, but it is still considered a foundation of Old English onomastics and can still be used with care.

As for meaning: an onomastician friend, Heather Rose Jones, pointed out that the essence of a name, what makes it a name as opposed to a descriptor, is that it is not a literal term, that its literal meaning is secondary or ignored. For example, she is neither a purple-flowered Eurasian heath, nor a prickly bush that typically bears fragrant flowers.

In Old English bithematic names, Prosopography of Anglo-Saxon England (PASE) lists Eadwulf 58 times, "prosperity wolf". Leofstan, 76, "dear stone". Frithwulf and variations, 3, "peace wolf". Wilburg, 1, "will fortress". Wigfrith, 16, "war peace". Ealdwine, 16, "old friend" (remember that this is an infant baptismal name).

Old English people did not have inherited surnames. If a family was using bithematic names, they tended to share a theme between generations. For example, Wessex's King Æthelwulf had children Æthelstan, Æthelswith, Æthelbald, Æthelberht, Æthelred, and Ælfrēd the Great.

So the Old English appeared to have chosen names for sound or for family reasons. "One from column A, one from column B" may be an appropriate metaphor.

I gather from Hall (q.v.) that one of the standard works is Fran Colman, The Grammar of Names in Anglo-Saxon England: The Linguistics and Culture of the Old English Onomasticon. While I have only glanced in it, I think his position can be inferred from the titles of chapter 3, "Names are not nouns", and chapter 4, "A name is a name".

As for elves in particular: I have little idea. Alaric T. P. Hall has written about it: his Ph.D. thesis from 2004 is The meanings of elf and elves in medieval England (online from the Univ. of Glasgow), and he used that as a basis for his 2007 book, Elves in Anglo-Saxon England: Matters of Belief, Health, Gender and Identity. Each nearly 300 pages. I looked at the onomastics section of his thesis, with

growing preference for certain fixed combinations which meant that by the eleventh century, dithematic names were generally of fixed form and often opaque as lexically meaningful compounds (Colman 1992, 55–67; cf. Clark 1992, 461; Kitson 2002, 105–6).

and

Names primarily denoted their bearers rather than being lexically meaningful compounds (Colman 1992, 12–16; cf. Barley 1974, 1–13), and Germanic names probably always included elements which were not transparently meaningful, either because they had been borrowed from other languages or because linguistic changes had rendered once-transparent elements obscure. Thus although it is clear from puns and literal translations that Old English dithematic personal names were potentially meaningful (see Robinson 1968, 35–57; 1993 [1970]; Harris 1982), it is considered unlikely that patterns in the pairings of elements in Old English names reflect the elements' lexical meanings.

Though he softens these statements a bit in his book.

He then points out that Ælf- is used only as a modifier, not standalone: "this implies that a name-bearer might be like an ælf, but never be an ælf himself." He also suggests that name elements appear to have positive connotations. He lists prothemes and notes that, of animate beings, people are most common, but there are also animals of high cultural prestige, and "god" and "elf", but monsters pretty much left the Old English naming stock early.

Although new elements were added to the system, such as Peoht- (‘Pict’) and Trum- (‘strong’, a Brittonic loan), and although some seem to have been dropped, such as -ides (‘lady’), several elements which had been lost from the common lexicon survived throughout the Old English period (for example, -fl{ae-}d, Tond-), presenting a real possibility that the presence of ælf in the personal name system merely reflects the semantics of a long-distant time.

The rest of the book looks like an interesting study.

Peter R. Kitson, "How Anglo-Saxon Personal Names Work", is a wonderfully lively article, though getting really deep into dense philology. (Oh, and he wrote, "Name-compounds are not, then, closely conditioned semantically"). Unfortunately, something went wrong with the PDF generation: boldfaced text, mostly names, got some characters mangled (I think it was those with a macron, _, on top). And a few pages are mangled. It also cuts off short, but downloading it and reading in Adobe Reader appears to help that.