Archive: early 17th century (in the sense ‘place where records are kept’): from French archives (plural), from Latin archiva, archia, from Greek arkheia ‘public records,’ from arkhē ‘government.’ The verb dates from the late 19th century.

Research: late 16th century: from obsolete French recerche (noun), recercher (verb), from Old French re- (expressing intensive force) + cerchier ‘to search.’

Index: late Middle English: from Latin index, indic- ‘forefinger, informer, sign,’ from in- ‘toward’ + a second element related to dicere ‘say’ or dicare ‘make known’; compare with indicate. The original sense ‘index finger’ (with which one points) came to mean ‘pointer’ (late 16th century), and figuratively something that serves to point to a fact or conclusion; hence a list of topics in a book (“pointing” to their location).