To start with, your question fits better with the history of science SE.
What hinted to physicists that a heavier electron like particle would exist?
Was it required as part of a theory or some common observation?
Nevertheless it has a nice core of physics in it. The answer is
- The discovery of the τ lepton was entirely unanticipated (in A Pais' words: Inward Bound, p 608) which emphasizes the heroism of Perl's discovery and the merit of the 1995 Nobel Prize. Much like the μ, "nobody ordered that". No hints, no theory undergirdings!
In fact, Perl's unique role in this is that he persisted, in the face of skepticism and jaundiced, crabbed adversity, and all but bet his reputation on something hardly anyone wanted, given the complications of D meson thresholds in Mark I (a fascinating experimental issue, albeit technical: the D and the τ differ by only 5% in mass!). In a dogged tour-de-force in 1974, he managed to convince the community. He details some of that in his 1995 Nobel Prize Lecture.
- The key point: two years before (an eternity at the state of flux of the time) Bouchiat, Iliopoulos and Meyer had figured out the anomaly cancelation matching of quarks and leptons generation-by-generation, required by the consistency of the gauge theory. With the discovery of the charmed quark, the first two generations were now complete and tightly fitting. No further quarks were apparent! So, finding a "just so" lepton was slightly unwelcome (in Pais' words, it "upset the balance"!)——but, of course, after its discovery, a 3rd generation was all but inevitable, and the discovery of the b and the t were more likely than not.
So, Perl damned the torpedoes and rubbed the blunt fact of the existence of the third generation (and the crucial Kobayashi-Maskawa discoveries relying on it!) on the dubious, unconvinced, community—my reason for using the "heroic" hyperbole above.
Theorists and phenomenologists, like Tsai, always work out the formal and experimental implications of every theoretical possibility, covering the waterfront; but the history in Wikipedia is somewhat misframed: an unmotivated "what if" speculation does not make an anticipation, and prophesy. This is a clueless debased notion of what a constitutes an anticipation and early credit. Sadly, a large number of hit-or-miss types made a profession of this.
By contrast, Pauli solved an actual problem with his neutrino hypothesis/anticipation; Lee and Gaillard with the anticipation of charm. But the τ did no such thing. It was an adventurous exploration of the type: "well, if you can have $e+~e^-\to \mu^+ \mu^-$, can you have something like
$e+~e^-\to \tau^+ \tau ^-$ on top of it?”
I recall Feynman was excited that an unpopular investigation such as this led to a brutal fact such as the τ which challenged theory at the time.