wow, UK-Nice by train!? It takes at least 10h!
Nice or Antibes (or Menton) should be great for accomodation, preferably near the train station. Monaco is well served, about 2 trains/hour on each direction. Maybe there is special trains during race weekend but I can't confirm...
Antibes is the farthest of these 3 town, it cost 7€ (one way) and 1h to join Monaco. But if you're at the window, landscapes are great, especially after Nice!
Nice is twice closer, and Menton is the closest but it is on the other side of Monaco (next to Italy).
If you're looking for fun, there's a lot of pubs on the Old Nice (which is just at the end of "Promenade des Anglais") and on the Old Antibes (between "Le Cap d'Antibes" and the main harbour). You could find lots of english people on these pubs.
For your mum, Monaco oceanographic museum is great, Menton is a beautiful city. If you want, near Antibes and Cannes there is Biot which is famous for its glassworks and Grasse which is capital of fragance. I know you can visit glassworks and perfumeries but I never try... Be careful, Grasse is a real mess, even for its citizens!
Cannes is next to Antibes and Festival de Cannes began May, 12th. Red carpet ceremony is famous but you have to come 3h before the start if you want to see something... But La Croisette is great to walk and there's some activities on it during the festival.
There's a lot of others things to do but I'm not a good guide. You should easily find websites which tells you what you must see
Just after Italian border, you could visit San Remo.
I can't help you for Paris, but things you must see are Eiffel Tower (of course), Les Champs Elysées, Place de la Concorde, Jardins des Tuileries, the Louvre, Notre Dame, Jardins du Luxembourg, les Invalides, Trocadero and Arche de la Défense. The easiest way to move in Paris is subway. Visit
http://www.ratp.fr for any information, it is the official website of Paris subway
@richard_leeds: thx for the link on Picasso Museum, I don't know there is one at Antibes until today!
ps: if you come to Nice, there is a ice-cream maker called "Fenocchio". He had 2 shops on Nice and his icecreams are awesome