In the LSJ entry, the semicolon is placed after the citation to Aristotle’s Atheniensium Respublica, indicating that the form μύστησιν belongs to the next citation, not to Aristotle,. The citation is “IG [Inscriptiones Graecae] 12.6.9,” an inscription. You can probably find it somewhere online. The dative plural form in -ηισι(ν) is Ionic or perhaps pseudo-Epic.
It’s not clear from the LSJ entry why the η in μυστησιν is not marked with an iota subscript while in the parallel form ἐπόπτῃσιν the η is so marked. An ancient inscription would not have used iota subscripts – a medieval convention; beginning sometime in the Hellenistic period, the iotas of long diphthongs ceased to be pronounced and eventually to be written, too, so the iota subscript of ἐπόπτῃσιν is probably just an editorial addition, like the breathing and accents.
Adding to the above by Hylander: μύστησιν is dative plural. In Classical Attic prose it would be μύσταις. Dative plurals in -ησι(ν) or -ασι(ν) are the older forms. You find them in inscriptions and in fossilized forms like Ἀθήνησι “at Athens” (locative). In Attic of the late 5th c. BCE the younger dative plural -αις took over. (For ἐπόπτῃσι, I suspect that the inscription in question reads ἐποπτηισι)
If you read inscriptions you can find for the 1st decl dative plural -αις, -ησι, -ασι (after ε, ι, ρ in Attic) -ηισι, or -αισι, among others (…isn’t Greek morphology fun?).
Unfortunately Perseus is an unreliable tool for morphology, especially anything to do with other dialects, inscriptions, poetry, etc.
If you look at the print version of the LSJ, you’ll see that the reference is actually to Inscriptiones Graeae Editio altera (the little superscript 2). I have trouble finding a print version, but it seems to get quoted from line 14 here:
Notice, however, that this section is a restoration. So does this form exist anywhere except in the reference books? Where does it actually come from?
The subscript iota of ἐπό]πτῃσιν is carved onto the stone, however. (As an adscript.) You can see Threatte for an account of the development, beginning on p.353. ηι seems to have started to show up as ει first, by about 300BC, a few decades after this inscription.
EDIT: @Hylander, can you post the page that you are referring to? It seems to be this alternate reconstruction of the same inscription (so again missing most of this section, if PHI is to be believed): https://inscriptions.packhum.org/text/232801
EDIT2: This is probably what Hylander is referencing, and is used for the reconstructions above, though “-εσιν/-εισιν” is still provided: https://inscriptions.packhum.org/text/346478
Just to clarify my comment above: -ησι without the iota after eta (i.e. -ῃσι or -ηισι) was the original 1st decl dative plural form. It was written -εσι in the older alphabet, as Hylander said. It is derived from the locative in PIE (fossilized in Attic forms like Ἀθήνησι). So in inscriptions you’ll find for the feminine dative plural definite article: τῆσι/τεσι, τῃσι/τεισι, ταῖς. In the late 5th c. BCE the -αις form largely supplanted the older -ησι and the secondary form -ηισι, becoming the standard ending in Classical Attic.
You’re getting this from Threatte, I assume? Could you give a page number for the dates? Anyway, this explains the form. The inscription from c. 475 shows forms like δραχμεσι and αυτεσι, and the μυστ[εσιν] restoration makes sense. But the later inscription quoted by the LSJ, with all of its -αις forms, should not have blindly used the earlier for a restoration.
EDIT: pg. 107 of Hylander’s link to Böckh, apparently, number 71, and it does match the second PHI entry above. Böckh shows two more letters of μυστ[εσιν] as visible than the later publication used by PHI.