How did this one particular family rise above everyone else? Where did their money come from? Their power?
Beyond the VIIth century, we know little of most of Frankish aristocratic family and that includes Peppinids and Arnulfians. A great deal of hypothesis comes down to their genealogical origins and relations to other families during and after the Carolingian takeover, which does let out a lot of non-Austrasian lineages.
Eventually, the Etichonid dynasty appears in text with Adalric-Ethico himself, native from the sub-kingdom of Burgundy and genealogically to the regions' great families and thus probably part of the regional aristocracy himself with the implication of land grants, honors and benefices from the royal palace. His possible ascendents (Amalgar and Aquilinia) in particular seems to have been patron of upper Burgundian religious establishment, an important institutional and economical display and entranchement of power, that Adalric and his descendents perpetuated in Alsace. The familial wealth of Etichonids in Burgundy might what have allowed Adalric to fund his rise to power in Alsace in first place.
The region was in the VIIth century was disputed by the Austrasian aristocracy, the Burgundian aristocracy and probably the local nobility in search of autonomy, who gained something out of the establishment of a duchy under Dagobert I to protect the Lower Rhine against transrhenan raids (but also to limit Austrasian hegemonism in the region) and ruled by appointed domestici.
We don't know ewactly why Adalric-Eticho was appointed by Childeric II : proable reasons are an award for Adalric's support of the king in Burgundy, a possible role into the murder of Leodegar, the necessity to deprive Austrasian nobility to control the region, etc. From these grounds, Aladric-Etichon effectively acted as a territorial prince, especially with the continuing collapse of Merovingian authority, as many peripheral regions did in this period (such as in Aquitaine, Provence or even Maine in the 700's) : percieving public revenues, control of public fisc and landed property, grant of public land and public honors, foundations of monasteries (altough, at first, on behalf of the Merovingian kings) etc.
These were basically the markings in the limits of his jurisdiction of an autonomous ruler that avoided being removed by Neustrian or Austrasian families alike by regularily switching sides, allying eventually to Peppinids, and carefully avoiding to outreach himself in bidding for further control and instead monopolizing political reorganisation in the region. The Life of St. Germanus in particular depict Adalric as "oppressive" and ready to bully or even crush resisting nobles whose properties were confiscated and redistributed; while other sources insist on his piety in the support or foundation of monasteries and "re-organization" of the region, probably with less-than-legally obtained goods and lands, whose donation or trust could thus be considered an early form of "laundring".
You might say Etichonids took control of Alsace thanks to a series of opportunist policies in relation to late Merovingian Francia, brutal policies in their new home turf, and a tradition of monastic/religious reshaping of Alsace