I have a question about a flight number of delayed flight. I think that if the daily flight(e.g. OZ9609) is delayed until the following day, there are two flights with a same flight number. In this case, I think operating two flights with same flight number could cause confusion to passengers and systems related with operating the airport. So I want to know there is any regulation that force to change a flight number if situation like this case occurs.
No, there's nothing that
forces a flight number change besides avoiding multiplicity
In general terms, it may be an unwanted complication to have two flights on the same route, which operate in the same day, with the same flight number. This is specially troublesome with delays in which the first flight ends up departing on a different date because of RPLs (Repetitive Flight Plans) may be on file with aviation authorities.
Take for example Delta's flight from Santiago, Chile (SCL) to Atlanta. Operates daily as DAL146 departing some time around 10pm local Santiago time and arriving into ATL early in the morning of the next day. There is a RPL filed with the Chilean aviation authority with the general flight info (daily operation, always same aircraft time and scheduled time of departure, generally the same route, and flight level, etc). This eventually gets overriden every day by the final operational flight plan that takes the most recent conditions into account, but for tactical reasons, the RPL is filed days or weeks in advance.
Now, consider what happens if today's DAL146 is delayed for whatever reason and is now scheduled to depart at 03am local time tomorrow (a 5 hour delay). IF the flight departed as DAL146, there would be a lot of reasons for confusion:
two flights departing the same city, same route and same flight number on the same day. This messes up a little bit with airport systems, air traffic control and the own airline's systems as the first(delayed) departure would cause the future flight to be seen as "departed on time" hours before its actual departure
For this reason, cases like this usually end up with the rescheduled flight getting a new flight number (you may see DAL146A, DAL146Z or even a completely new flight number, depending on internal airline's protocols) so even if the two "identical" flights end up departing on the same date, there would be no confusion.
Another reason for this to happen is multi-leg (from A to B to C) flights in which one leg (A to B) is delayed but the next leg (from B to C) can be operated by another aircraft in order to minimize passenger disruption and help with the fleet positioning. Cases like these, if left unhandled, could lead to more than one flight using the same flight number at the exact same time, which is something that simply cannot be had, so one of the flights will get an alternative flight number to prevent the conflict
An airline's flight number for commercial purposes is not necessarily the same as the identifier the aircraft uses for air traffic control purposes. For example, British Airways flight may show up on a flight tracking web service as BA### but air traffic control identifies them as "Speedbird ####" where the number may or may not be the same as the airline's commercial flight number.
So, to answer your question, there is no FAA regulation that forces the airline to change commercial identifiers. The ATC identifiers are set when the dispatcher files the flight plan, and they must be unique to ATC. The system will reject it if the identifier is in use, or will be during any period of flight.