The few forecasts so far on this question are quite divergent: 0%, 0%, 1%, 70%, 75%.
I think it's useful to look back at how long previous prime ministers stayed in office. Going back to Romano Prodi, who took office on May 17, 2006, here are the numbers:
Romano Prodi, 1 yr 357 d
Silvio Berlusconi, 3 yrs 192 d
Mario Monti, 1 yr 163 d
Enrico Letta, 300 d
Matteo Renzi, 2 yr 294 d
Paolo Gentiloni, 1 yr 171 d
Giuseppe Conte, 2 yr 257 d
Mario Draghi, 1 yr 251 d
Giorgia Meloni, 62 d so far, since taking office Oct 22, 2022.
Only 1 out of the 8 prime ministers immediately before Meloni served less than 1 year, and Oct 1, 2023 would be just under 1 year. Looking at nothing else, I'd put the chance that she'd be out of office before Oct 1, 2023 at ~13%. But Meloni came in with a clear voter mandate, so I think that makes her chances of staying >1 year even greater. That said, she is PM during unusually difficult times, with unusually high inflation, and that would be a force that could shorten her tenure. However, difficult times often push politics to the right (which is probably how she got where she is now), and she represents the right, so perhaps difficult times won't hurt her. Also, any scandal could push her out, but I haven't heard of any such possibilities.
So, I think the chance she'll still be PM on Oct 1, 2023 is really high.