1 month stay recommendation in Bern? (family of 3)

Switzerland is very expensive, and Bern is probably higher than average because it's the capital. I'm afraid a budget of CHF 60 per night is extremely low. You can take a look at booking.com, which has listings for apartments as well as hotels, to see some options, but I think they will all be way over your budget. If you use the Map view and zoom out, you'll see some less-expensive lodgings as you get farther from Bern.

I've never used Airbnb myself, but that would be another place to look.

Is there a reason you’re choosing Bern?

I have found “reasonably” priced places (of course “reasonable” is subjective) on Sabbatical Homes. Here is the current list for all of Switzerland. https://www.sabbaticalhomes.com/search?listingtype=HomeForRent&propertytypes=0&typeids=1&lat=46.818188&lng=8.227512&lathome=0&lnghome=0&radius=10&placeidsearch=ChIJYW1Zb-9kjEcRFXvLDxG1Vlw&exactdates=True&includeanydestinations=False&includenonsimultaneousexchange=False&expectedstay=0&expectedstayunit=0&includenearbycities=False&amenities=0&surroundings=0&userstatus=0&youngkids=False&smallchildrensafe=False&smoking=False&nosmoking=False&pets=False&car=False&people=0&housesittingpets=False&housesittingplants=False&housesittingmail=False&housesittingstrictlynone=False&accomodates=0&flexibledates=False&offeredbyacademic=False&offeredbyindependentscholar=False&exchangesinterestedinmyareaonly=False&hideadvancedoptions=False&searchtype=SearchPage&totalcount=23&PagingInfo.totalrecords=23&PagingInfo.recordsperpage=50&PagingInfo.pageindex=0&PagingInfo.lastpageindex=1&searchtext=Switzerland

Nothing in Bern but perhaps you can be flexible?

Thanks Laurel!
This is the most useful option - especially, again, because we plan to be staying long term (at least a month).
Much appreciated.

if you will be in or near Bern with a 10 year old you may want to become familiar with the Gurten. https://www.gurtenpark.ch/en

It is Bern's own mountain, right in town, trans take you to the Gurtenbahn, a funicular, which run you up to the top. All included in your Bern Mobile ticket or pass. Up there are all sorts of fun things to do for kiddoes, including a ride on railway, a luge/toboggan (only CHF4) and good playground. Lots to do for adults too, including a restaurant, nice walks, bikes...

I love it up there.