Venice in Late October or Early November with 14 year old

A quick search on https://www.booking.com with some random criteria found 236 hits, but the USD limit is 216 USD not 200, so some of those will be over you limit. 121 if the limit is 162 USD. (The limits are planned in Euros)

If you want quiet, I would look in Cannaregio or Castello.

Also check https://europeforvisitors.com/venice/venice-hotels.htm

I wouldn't wait too long booking. Venice is popular.

I don't recommend staying on Burano as this is your first trip to Venice. You will spend money and TIME going to and from Venice when you arrive, when you want to see Venice, and when you leave. Stay in Venice and maybe take a day trip to Murano, Torcello, and Burano. Buy a day pass for the vaporetto (for that day only if you want to save money or for the whole period, if you want to save energy)

Take a look at Hotel al Ponte Mocenigo. You can find lots of photos and info about it on booking.com and on its own website. It is only steps away (no bridge to cross with your luggage!) from the San Stae vaporetto stop, only 3 stops from the train station, yet an easy stroll to the Rialto Bridge, Rialto market, and much more. It's also on a quiet street, away from the hustle and bustle yet close to some wonderful cafes and shops. It's recommended by a couple of people in this very recent discussion: https://community.ricksteves.com/travel-forum/italy-reviews/2-or-3-star-hotels-in-venice-cinque-terra-florence-sienna-rome-and-amalfi. I agree: DON'T stay on Burano. You'll spend an hour just going to or from that island, and once you get off the vaporetto in the main part of Venice, you will still be far away from much of what you want to see and do in Venice.

You'll probably get more responses if you post in the ITALY DESTINATION section; this section is for reviews of venues already visited.
Have a great trip!

Best to post this in the Italy section - but don't stay in Burano - you can easily take 1/2 day trip there. Stay in Venice -so much to see and do there.

We really like Pensione Guerrato and have stayed there several times. The location is very close to a vaporetto stop, but far enough away from Rialto Bridge to not be in the mob scene. It does have some stairs to climb if that is a consideration. If you book there let us know and we can give you directions that will eliminate a lot of walking with the luggage.

I found a site called "Monastery Stays". There are several in Venice. The prices were per person and are reasonable. Probably more like a hostel on low end or pensione on high end...but presume clean. Most of the listings are in historic buildings right in actual Venice, so would be very convenient.

I would suggest staying on Giudecca island. It’s a quick vaporetto trip over to St. Marks (15 minutes). Easily accessible from the train station, with only one bridge to cross from the vaporetto stop. Away from the hustle and bustle, and quiet. We stayed at this apartment in June 2018 with our two kids:

https://abnb.me/8WgyO602g4

There are several apartments available. The host is wonderful.

I would advise against staying on Burano. Burano is a 40 minute trip by waterbus to Venice proper, and even then, a further 25 minute walk to the centre of Venice, the Piazza San Marco.

The suggestion of Giudecca is good, but you have to use the water bus (aka vaporetto) to get to central Venice. We have rented an apartment on Giudecca for our next visit for a dozen nights in October. Giudecca is much less touristed than more central Venice, and so somewhat cheaper.

It is worth buying a vaporetto pass for the duration of your stay.

FYI your 14 year old grandson will qualify for a Rolling Venice transportation card. You can buy unlimited travel for 3+ days, 22€ instead of 40€. Just show his passport at the ticket office in front of the train station when you buy your transportation ticket. I had our granddaughters take a small school sized photo for the card but they didn’t need it. http://www.veneziaunica.it/en/content/rolling-venice

We stayed in an apartment (3br/2bath) in Cannaregio last summer and enjoyed the more quiet atmosphere. It was less than 200€ per night on HomeAway.

I echo staying at the Hotel al Ponte Mocenigo. I have two nights booked over Easter and keep checking back to see the latest Coronavirus status. I stayed there 5 years ago with my 13 year old daughter. A few steps away from the St Ste Stop which is the second or third stop on the Aliliuguna in a quiet location, and about a 5 minute walk to the Rialto Bridge but in a quiet area. If you are leaving on the train after that, it is a 20 minute walk to the train or bus if you pack light. The staff there is lovely, the breakfast room very quaint. I splurged and booked the room with the balcony. It was wonderful and I think around your budget at the time. This trip I booked late and am in a annex room but those were nice as well.

A few things I did with my daughter that she enjoyed. The first is we experienced Opera at the Musica a Palazzo. It is in a stately home and you are a part of the experience moving from room to room with the singers. Also there are great free walking tours, and of course the standard tourist sights. Enjoy!

I've stayed twice at the Santa Margherita Guest House in the Dorsoduro area. Clean, quiet, modern bathrooms, definitely budget. I wouldn't call it charming, but it's a good, quiet location, 5 minutes' walk to the Giudecca and Grand Canal ferry stops. 15-20 minutes' walk to San Marco. 3 nights in Venice is infinitely better than 2.