Hi OP, I know how it feels when that depression comes sweeping in, it really is like an engulfing wave of cold dark seawater ironically. Sometimes just toughing it out and waiting for it to lift a bit is the only option. I also know just how hard it is to do anything when you feel like that, let alone be bright fun happy mummy for your child. I sometimes have to hide it so that my kids don't experience my depression, and it feels forced and fake sometimes, like I am being too bright, too loud, and that they will see through it.
They won't see through it. They will see mummy smiling, and take that at face value. It's the wonderful thing about kids. You don't need to do much to keep them entertained, as pp have said, let them scout about on the beach, get wet and muddy, explore some new places, and get plenty of fresh air, even cold and wet air. They will sleep brilliantly after a day at the seaside, and will have lovely memories. The fresh air and seeing your child enjoying themself will also do you good too. Take it a day at a time, pick something simple to do each day, try and get plenty of outdoor time, and don't make it too hard for yourself. Try and factor in some relaxation time, consider downloading some audio books or podcasts perhaps, for in the evenings after your child has gone to bed, to give you some time for yourself.
One of my best holiday memories is of a day trip we took to Blackpool. It rained literally the whole day, and my parents were gutted, they'd saved for ages to take us on this coach trip. We loved it, we went on the penny drop machines on the arcade, ate junk food, got soaked on the beach, went to Madame Tussauds (dad took me into the chamber of horrors, mum wasn't impressed but it was awesome!!), and we even saw the donkeys. It was utterly wonderful as far as seven year old me was concerned. I think we all slept all the way back on the coach!
You can do this lovely, I know it feels impossible right now, and like the weight of the world is on you. But keep putting one foot in front of the other and you will come out from under the cloud. You really will. If you find yourself going through hell, just keep going.