To begin with, it might be a Circum-Baltic areal feature we could call marked predicative construction (as opposed to the unmarked/less marked nominative/accusative), as the following source suggests: On the marking of predicate nominals in Baltic. Whether the Slavic languages were the first ones to acquire this construction, which later spread further, I'm not sure, but here's an idea that has just occured to me:
As the instrumental is (at least in Czech) used to mark the agent of a passive clause, such as...
Jan byl zabit vrah-em.
Jan.NOM was killed murderer-INST.
"Jan was killed by a murderer." (INST obligatory)
...I'm beginning to wonder whether the predicative instrumental construction could have arisen through a shift in agentivity - of sorts. I'm only trying to catch a glimpse of something that's just evading my mind now, really, but the notion of temporariness or changeability in the instrumental constructions as opposed to a relatively greater stability in the nominative constructions also seems to evoke the notion of dynamicity as opposed to stativity in a way. A look at the following examples in Czech might elucidate what I mean:
1a. Učinil-i Jen-a.
"They made Jan."
1b. Učinil-i učitel-e.
"They made a teacher."
2. Učinil-i Jen-a učitel-em.
Made-3PL Jan.ACC teacher-INST
"They made Jan a teacher."
3. Učinil-i z Jen-a učitel-e.
Made-3PL from Jan.GEN teacher-ACC.
"They made a teacher from Jan."
Cf. the passive:
4a. Jan byl učiněn [jimi].
Jan.NOM was made [they.INST].
"Jan was made [by them]."
4b. Učitel byl učiněn [jimi].
Teacher.NOM was made [they.INST].
"The/A teacher was made [by them]."
5. Jan byl učiněn učitel-em [jimi].
Jan.NOM was made teacher.INST [they.INST] (potential ambiguity)
"Jan was made a teacher [by them]."
6a. Jan je učitel-em.
Jan.NOM is teacher.INST.
"Jan is a teacher (now, perhaps temporarily, maybe not for real)."
6b. Jan se stal učitel-em.
Jan.NOM became teacher.INST.
"Jan became a teacher." (INST required, irreplaceable by NOM here)
7a. Z Jen-a je učitel.
From Jan.GEN is teacher.NOM.
7b. Z Jen-a se stal učitel.
From Jan.GEN became teacher.NOM
Hence, I'm picturing something like the following:
Jan's been made a teacher-INST. → Jan is a teacher-INST.
As if a non-subject agent (or the more agentive-ish non-subject), unless made oblique by an intervening preposition, required the instrumental by default, originally, perhaps, in the passive only (?), then (?) in other contexts too?
The agentivity shift (if that's the right way to call it) is just a spontaneously materialized idea, really, but somehow I can't stop feeling that the two phenomena are connected in one way or another. I'll definitely keep thinking about this issue.