Given LHR is way out west of London and LCY is to the east, you'll have central London in between.

Since you (i.e., everyone except your mum) arrive before she will and you'll all leave from LHR, I'll suggest you take all the luggage (including hers, so she won't have anything to carry), check yourselves into a hotel. Once you checked in, go to your pre-arranged meeting point in London. Not much point of you going to LCY to meet her though, but you can do it if you want to.

In terms of hotel, and given your timing, I don't think it'll matter much whether it's near LHR or not. You may want to stay around LHR so you don't have to take everything into London then back to LHR 24 hours later. No matter where you stay, you'll need to ask for an early check-in, or at least ask to leave your luggage at the hotel.

Regarding meeting point, again, it's up to you. To reduce stress for your mum, you can arrange to meet at Liverpool Street Station. There's a bus from LCY that takes you straight there. You can also consider the Dockland Light Railways (DLR) for her. That can take her to Tower Gateway where Tower Bridge and the Tower of London are nearby. Even simplier is for her to take a taxi straight to wherever. For reference, LCY to Tower Bridge is around £10.

Hope this helps.