for your friend and also for you for being there for her.
In the first instance, your friend should forget about any notion of getting into debt or otherwise paying for her DMs funeral if the DM didn't have any money and the daughter doesn't either.
Sadly I've heard about well meaning relatives saying things like 'we 'had' to take out a loan we couldn't afford/go without other essentials in order to afford a funeral'. While people might want to give people a good send off, there's no requirement to pay for other people's funeral even if it's a parent or other close relative.
The liability stops with the estate of the deceased person and if there is no money available, then the local authority is legally required to provide a basic funeral. A silver lining in this particular cloud is that COVID restrictions mean that no-one is expecting a 'proper' funeral, catered wake etc, so this will release the pressure somewhat.
Do you know if your friend was on the tenancy? If so, she might be liable for the rent from the date of death, so she probably needs to update her benefits claim to see if she is entitled to some/more HB.
You say it's a private rental, not LA/HA type, but in any case, it's probably worth her writing to the landlord to say that her DM has passed away and ask for a short period of 3 months to get everything sorted so they don't think they've just stopped paying the rent and start aggressive eviction tactics. Another thing for her to think about is whether she wants to stay in the property or move somewhere else.
She needs to review her DMs affairs to see if there is any inheritence due - if there's little or no assets and no complicating factors, she won't need to apply for probate, but if she has no income, she needs to see if there will be any money due that may affect her benefits claim. She will also need to update her benefits claim as she will no longer be entitled to carers allowance. I don't know if there is any sort of berevment support, which will allow it to continue for a short period, before she is treated as a standard unemployed person.
Moneysavingexpert has a 'what to do when someone dies' guide, which will provide her with a useful checklist to work through.
www.moneysavingexpert.com/family/what-to-do-when-someone-dies/