To confirm what Relaxed said and detail a bit more on your specific situation, there are a couple interesting elements.
The conditions of the SNCF specifically mention that the ticket, included the e-ticket/printed ticket, is valid for the same day after validation or after the indicated train time (in the case of e-ticket) (paragraph 3.3):
Après le compostage, le billet doit être utilisé pour un départ le jour même et le trajet doit être terminé dans les 24h qui suivent le compostage ou, dans le cas d’un E-billet ou d’un Billet Imprimé, dans les 24 heures suivant la date et l’heure de départ du train.
It also mentions in the following paragraph (3.4) that in case of a paper ticket (not printed at home), the ticket should be validated again in case of a voluntary stop:
En cas d’arrêt volontaire en cours de trajet (dans les limites reprises à l’article 3.3 des Tarifs Voyageurs), le billet Papier IATA, le billet Papier ISO, le billet électronique ou le Billet à Valeur correspondant à celui-ci doit être composté à nouveau au départ de la gare d’arrêt.
However, to access a train, there are limitations. Some trains (all TGVs, some Intercités) require a seat reservation, and you cannot get on the train if you haven any. In France, it is not offered to buy a seat reservation without a ticket (except for monthly/yearly pass holders and personnel).
In your specific situation, the trains serving Nice, Baulieu-sur-Mer and Eze are regional trains, not requiring any reservation. This means with a TER ticket, you can access any train (on the day of your reservation, after the validation/printed time). You have multiple options to book a train ticket on this itinerary:
buy one ticket for Nice to Eze and you stop on the way. According to voyages-sncf.com, it would cost you 2.80 euro (as of May 2015). You can buy it online or at the station (vending machines accept credit cards and coins). Buying at the station offers you more flexibility on the date of travel.
buy two tickets, one for Nice - Beaulieu-sur-Mer and one for Beaulieu-sur-Mer - Eze. The price is higher (2.20 euro and 1.40 euro). The advantage is if you plan to do it in two days, but I doubt it is the case. You can buy them online or at the station.
One thing to know is that the payment system of the SNCF does not allow one credit card to be used more than 3 times within 24 hours (e.g. if you buy Nice/Beaulieu and Beaulieu/Eze, you only have one transaction left for the rest of the 24 hours).