Which neighborhood in Florence?

Try to stay near the Duomo. It really is the hub of the city. The Hotel Pendini at Piazza della Republica is very nice. Elevator access, good breakfast, large rooms. For transportation from the train station, take a taxi. While you might easily get around without luggage, that to-and-from the train station is a pain.

The areas you and your mom will want to visit are within a fairly compact area, but I wouldn't say it is walkable for an 86 year old woman in the heat of June. Florence does have a good local bus system. See http://www.visitflorence.com/moving-around-florence/by-bus.html for information, and from there you can also link to a map (http://www.ataf.net/System/files/mappe/ESEC%20mappa110x100.pdf) showing the bus routes. On this map you can see where the train station is.

You'll be spending most of your time in the area between the train station and Santa Croce, and mostly on the north side of the river, though there are some great sites across the river as well, and some slightly north of the Duomo. This is mostly level walking, but I'd suggest familiarizing yourself with the buses.

if you stay in that general area, you should be able to get around okay. Your airbnb host should be able and willing to help you figure out the buses. Look for someone with good reviews that talk about helpful hosts. Also, make sure you ask about stairs.

There are also taxis and I believe there are pedicabs as well, so I'm guessing you will have no problem getting around.

Enjoy!

In your shoes I would rent a wheel chair. That is what we did when we took my mother in law to Barcelona, where my brother in law was living at the time.

The historical center of Florence is largely flat, so it shouldn't be too hard to push her around. Distances in Florence are too small to bother with the taxis or buses, besides inside the historical center only electric buses can operate and on limited routes and with limited frequency. I'm sure you won't want to bother to get to a bus stop or a taxi stand when the next sight is only 500 ft away, and often the bus stop/taxi stand might be farther than the next sight you want to visit, however at the end of the day all those small distances add up.

The City of Florence Tourist Office rents for free up to a max of 3 days (they just want a deposit of €150). There is also the Ambulance service that offers them for free to tourists. And there are a couple of stores that will rent them at a fee and will actually deliver them to your hotel/apt and pick them up.

This is the info you need.
Free wheel chair rental at the station:
Tourist Infopoint
Piazza Stazione 4 - Tel +39 055 212245,
Email: [email protected].

The ambulance service offers the same free rental:
Misericordia di Firenze
Tel +39 055 417761

You can rent them for a fee at the following stores:
Dei Ortopedia - Via Pisana 71r – tel +39 055 224090
Galli Alessandro – Via Bronzino 157 – tel +39 055 704149
These two stores will deliver it to your place (or anywhere) and pick it up at the end of the rental.

In terms of neighborhood, anything within less than 1 km radius from the Duomo will contain all major sights.
Use this sight to draw a circle around a point: https://www.freemaptools.com/radius-around-point.htm
Enter Piazza Duomo, Florence, Italy in the search box, set the radius at 1 km (or even just 0.5 miles) and click on the Duomo (which is called "Santa Maria del Fiore" cathedral).