Etymology
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interim (n.)
"time intervening," 1560s, from Latin interim (adv.) "in the meantime, meanwhile," originally "in the midst of that," from inter "between" (from PIE *enter "between, among," comparative of root *en "in") + im, ancient adverb from stem of pronoun is "this, that." As an adjective from c. 1600.
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*en 
Proto-Indo-European root meaning "in."

It forms all or part of: and; atoll; dysentery; embargo; embarrass; embryo; empire; employ; en- (1) "in; into;" en- (2) "near, at, in, on, within;" enclave; endo-; enema; engine; enoptomancy; enter; enteric; enteritis; entero-; entice; ento-; entrails; envoy; envy; episode; esoteric; imbroglio; immolate; immure; impede; impend; impetus; important; impostor; impresario; impromptu; in; in- (2) "into, in, on, upon;" inchoate; incite; increase; inculcate; incumbent; industry; indigence; inflict; ingenuous; ingest; inly; inmost; inn; innate; inner; innuendo; inoculate; insignia; instant; intaglio; inter-; interim; interior; intern; internal; intestine; intimate (adj.) "closely acquainted, very familiar;" intra-; intricate; intrinsic; intro-; introduce; introduction; introit; introspect; invert; mesentery.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit antara- "interior;" Greek en "in," eis "into," endon "within;" Latin in "in, into," intro "inward," intra "inside, within;" Old Irish in, Welsh yn, Old Church Slavonic on-, Old English in "in, into," inne "within, inside."
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meantime (n.)

also mean time, mid-14c., mene-time, "interim, interval between one specified time and another" (now only in in the mean time), from mean (adj.2) "middle, intermediate" + time (n.). Late 14c. as an adverb, "during the interval (between one specified time and another)." As a noun, properly written as two words but commonly as one, after the adverb. In the mean space "meanwhile" was in use 16c.-18c.

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interval (n.)
early 14c., "time elapsed between two actions or events," from Old French intervalle "interval, interim" (14c.), earlier entreval (13c.) and directly from Late Latin intervallum "a space between, an interval of time, a distance," originally "space between palisades or ramparts" [OED], from inter "between" (see inter-) + vallum "rampart, palisade, wall," which is apparently a collective form of vallus "stake," from PIE *walso- "a post" (see wall (n.)).

Metaphoric sense of "gap in time" also was in Latin. From c. 1400 in English as "a pause, an interruption in a state or activity." Musical sense "difference in pitch between two tones" is from c. 1600. Related: Intervallic.
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