Hi there
Our parents separated when we were very young (I have a brother), so all trips were one parent, two kids.
We stayed in the UK, went to somewhere by a beach (for us it was newcastle, it has so many beaches, plus rock pools, lighthouses, the city, a boat across the river, the metro to get around, regular kids fun things like museums set up for kids, parks, playing on the beach, all an adventure because we were on a trip!). I have the most amazing memories, some of my favourite from my childhood. We were always on an incredibly tight budget, but we as children never felt it at all.
Full disclosure, now I'm Miss Jetsetter, I've travelled, lived and worked all over the world, - hello from Paris and I think France just scored a goal because there is massive cheering outside! - I love the weather and the cultures and so on when I travel, but I didn't get on a plane until I was 20 and as a child and even a teenager, we didn't need or want that part of it! At five, definitely not. Adventure is adventure for kids, the same things that make them happy will make them happy anywhere.
So maybe choices closer to home where you can find budget appropriate things. As I said, we took the train for a couple of hours, family railcard, had adventures doing things kids love, happiest childhood memories I have.
If you want to go abroad, really, take out the travel agent! A five year old as an adult, ridiculous, they charge massively for packages that are so simple to put together yourself, I'm always shocked that people don't realise this!
Find your own flights, easy jet have so many and are good prices as others have mentioned, they are actually really good. For somewhere to stay, it's easy to check out airbnb, same price no matter how many people, I have friends who went around the world with three young kids and were on a strict budget, they found family friendly places absolutely everywhere.
Get a guide book or look a bit online and find things your child will enjoy and you will enjoy! The Rough Guide guide books are full of so much information about absolutely everything, they are so good for making the most of a trip, plus how to manage money and costs is a big feature in them because they are for people who love to travel so they discuss backpacker budgets and all budgets.
It can be way more fun for you that way too, you can choose things you know you will enjoy together:)
Good luck!