An incorrect name or passport number on a Rinjani permit is a real problem, because the park’s FAQ states that the named trekker and ticket data cannot be changed once the ticket is issued. There is no publicly documented self-service typo-correction process, so do not assume a quick edit is possible. What you do in the first hours after noticing the error decides how much of your trek is at risk.
Contact the operator who booked the ticket immediately, with your booking identifier and a clear copy of the correct passport biodata page. Ask them to check with the park administrator whether an administrative correction is possible for that specific error. Some errors may be fixable with administrator intervention, but nobody can promise it in advance, and a correction that looks trivial to you may still require the park office to act on the booking record.
If the error cannot be corrected, the practical fallback is cancellation and a correctly issued replacement ticket, subject to availability for your gate and date and any applicable charges. Do not rely on a verbal reassurance at the gate, and never send a different person’s passport hoping staff will not notice, because gate staff compare the QR code against identity documents. Note that the date itself is treated differently from the name: the park allows one rescheduling for the same registered trekkers, requested by the day before the trek and subject to quota.
Prevention is far cheaper than any of this. Before paying, have the operator read back the passport spelling, passport number, trekking date, and entry and exit gates, and check them against the actual passport, not your memory of it. We do this read-back with every guest as standard practice, precisely because a single wrong digit in a passport number is the most common registration error we see.
The honest summary: identity errors are sometimes recoverable, sometimes not, and the outcome depends on the park administrator and remaining quota. Register carefully, verify before payment, and if an error slips through, move on it the same day, not the night before your trek.

