Can anyone tell me how to get from Rome to Nice as quickly and cheaply as possible? Is flying the only option?

Flying the ONLY option? Well, of course, not... though it may be the fastest and cheapest. Naturally, there can be only one "winner" in either category - - although walking, swimming and bicycling would perhaps tie for cheapest if you don't count any of the "personal" costs.

The website www.whichbudget.com will show you that blu-express.com offers air service. They're new, as far as I know (intra-Europe low-cost air travel seems to spawn a new airline almost every week!) so you may not find anyone here who has actually traveled with them.

And of course, train, bus and rental car are all options as well.

I haven't bothered to find any commercial sailing lines for this route; not sure if they exist.

Best wishes,

Rex

There are no routes by sea, but there is an overnight train which will waste the least time.

<<...there is an overnight train which will waste the least time...>>

This is true, if you don't mind

...dealing with your luggage from the time you check out of your hotel until the departure at 23:10

... the twenty stops it makes during the 10 hr 35 min trip... there is one period of 71 minutes, and another of 81 minutes with no stop.

...arriving in Nice at 09:45 having had no shower, 2-5 hours before you can check into a hotel room.

Different strokes for different folks.

www.blu-express.com flies you there for 9.99 Euros plus taxes and fees etc. etc.

Way cheaper than a night train even if you spring for the poshest sleeper cabin - you may not sleep much...

Rex has obviously never thought of leaving luggage at a hotel and picking it up later (even cheap hotels do this) or of putting it in the left luggage at the station.
When you arrive in the morning, you can leave your luggage at your hotel and may well find a room ready.
There have been other posts on this topic. Do hotels in the US really have rigid check-in and check-out times and not allow guests to leave luggage?

Yes, I have thought of that... I file that under "dealing with it"...

And yes, hotels will hold luggage... but they won't allow you access to the room for a shower, change clothes, whatever other personal activities might be desired...

I have previously stated my strong bias for a bed that doesn't disappear whenr you arrive at the station... and a (full) bathroom there in the morning, when I want to use it...