Hi TillyBob
There are two hoho companies in the city, http://www.barcelonabusturistic.cat/en/home and http://www.barcelonacitytour.cat/en/ and http://www.barcelona-tourist-guide.com/en/tour/barcelona-bus-turistic.html to explain how hoho buses work -in case you're not familiar with the concept. I personally like the red route from the first company. I do believe that if you purchase the tickets online you get a discount, at least in the first company which is run by the City Council and the Tourism Board: http://bit.ly/19DLNlX I guess the second one, privately run, must have a similar discount in their webpage as they are both neck to neck.
Barcelona is an all year round tourist destination but note though that if you're coming anytime between mid Jun to Mid August the hoho buses, both companies, are particularly busy (and I mean busy boy!) so it might not be so easy (and convenient for that matter) to use them. Despite there are quite a lot of them I've seen endless queues of tourists patiently waiting under the sun in many stops.
If your mother is not much for walking (sad because Barcelona is a city made for strolling!), note that each company has several routes (3 for the first one, 2 for the second) and with a day ticket you can use all of the routes of that company, but they're not interchangeable of course. Each route is normally 2 to 3 hours.