What is the name of the arched, white area, underneath the fingernail, by the cuticle (most obvious on the thumb)?

Lunula: from Latin, crescent-shaped ornament, diminutive of luna or moon.

The lunula (plural = lunulae) is a small crescent-shaped structure or marking, especially the white area at the base of a fingernail that Resembles a half-moon. The lunula appears by week 14 of development and has unique histologic features. The lunula defines the free edge of the distal nail plate. Lunular anomalies include changes in form and structure and in color.

Its normal color is due to the thick, opaque layer of partially kertanized matrix cells in this region. The lunula can be larger than normal (macrolunula), smaller than normal (microlunula) or absent (anolunula). It can also be nonconvex! The lunula can be confluent or spotted or can be characterized by longitudinal colored bands that traverse it. Alterations in the morphologic features or color (or both) of the lunula can be an indication of either a cutaneous or a systemic disorder.

References:

Cohen, PR: The lunula. J Am Acad Dermatol. 1996 June;34 (6):943-53; Department of Dermatology, University of Texas-Houston Medical School 77030, USA