check flight prices:
I try to buy a ticket for the 14th of july, and the price for a flight rome-munich is 60 eur, including taxes. It's good.
HLX is better: there's an offer for the 14th of luly (probably also for other days)
: venice munich 11 eur taxes included
they have offers if you reserve in particular days, check their web site.
in any case your mother can decide to start her itinerary from north to south or from south to north.
She can reach every city by train, slow trains are really cheap, fast trains (IC, ES) are a littlle bit expensive, but less than in other EU countries.
lenght of journeys:
venice- florence approx 4 hours by train
florence -siena 1 hour by bus or 2 hours by train
siena -s gimignano 45min -1 hour by bus
florence- pisa 1 hour by train
pisa -lucca 20-30 min by train
pisa -5 terre 1 hours 30min by train
florence-rome 1hours 30min-2hours 30min depending on the train
pisa roma 3 hours 15min -4 hours 30min depending on the train
in trenitalia web site you can check both timetable and prices, and reserve. You can't probably buy tickets for some local trains, you can buy directly at the station.
as for accommodations: all the cities you've mentioned are really expensive, and it's true that without a car is not advisable to find an accommodation far from the city centre (maybe in an area not well connected by public transports): try last minute offers, but expect to spend at least 50 eur for a single room per night, if your mother doesn't like to sleep in a hostel.
(single rooms are more expensive than double)