じ(ーーーー...) seems to be a phenomime, a word which evokes the idea of the state 'staring'.
じっ‐と is an adverb meaning something along the lines of being still, quiet, focused at somebody or something, something like 'stare', in some sense.
Then, it's not far fetched to suspect that these both words could be related, is one a contraction of the other, or, maybe has an adverbial と been added to じ in some way. Is there any connection?