Derivation from the びく(biku) root found in other terms such as びくっと(bikutto, “startedly”), びくつく(bikutsuku, “to shiver, as from fright; to startle; to become nervous”). The underlying meaning of the root appears to be twitch or start(noun).
This is one of many adverbs formed by the addition of the suffix り(ri) to the root, and then gemination of the middle consonant. The suffix may derive from the auxiliary verb り(ri), itself from classical verb あり(ari, “is, are”).
*biku → bikuri → bikkuri
The kanji spellings for this term are very rare. The kana spellings びっくり or ビックリ are much more common.