The basics are
1) A good sized cage, minimum 70cm by 40cm continuous floor space - not too high, so they can't hurt themselves if they fall.
2) A wheel that is the right size - 6.5" to 8" for a dwarf hamster, 11" to 12" for a Syrian Hamster (so cage needs to be tall enough to fit that).
3) Plenty of substrate, at least 3 or 4 inches deep - there are number of options now, not just pine shavings. Fitch paper substrate and hemp are popular non-wood options.
4) Torn up strips of white toilet paper for nesting.
5) Something to keep their nails and teeth worn down - a chew for teeth and a stone or piece of granite or similar under a water bottle or food bowl can help.
6) Water bottle (two is good in case one stops working or drains itself dry) and food dish.
7) A suitable house for building a nest - some hamsters will build it under substrate or elsewhere, but if you give them a good sized house/nesting box that is dark inside they are very likely to use it because it's like an underground burrow and darkness is important for them during the day.
So those are the basics, but more than that is 'enrichment'. This is basically enriching the hamster's environment so they can have least stress, develop more normal behaviours and do hamstery things like digging tunnels or have favourite places to go and safe places to hide in their environment. Enrichment generally means tubes, toys, etc and substrate to dig in. People who use tanks and bin cages are able to give 20 to 30cm deep substrate which means hamsters can dig a network of tunnels. That's not so easy to do in a cage but you can still manage about 15cm deep and that encourages them to dig as well. Other forms of enrichment are scattering some of their food so they can forage (I put half in the food bowl and scatter the other half), added variety of diet with a variety fresh veg (tiny amount - thumbnail size) and other supplements to their diet. Healthy treats - I get the Rosewood herbal chew sticks which are sugar free (hamsters can get dental decay like humans and dwarf hamsters are prone to diabetes) and this helps wear their teeth down as well.
Habits are part of enrichment too - so you can reduce the amount and extent of cage cleaning by having deeper substrate and spot cleaning, and using a potty litter tray, which you clean out every 5 days or so and the rest of the cage stays cleaner, so less stress for the hamster (full cage cleans are stressful for hamsters and partial ones are better). That's a whole topic on its own!
I wouldn't use a toy house but I know what you mean - some of the little colourful plastic ones look like toy houses. Wood or cardboard is better than plastic as Cypher says. I also think a house with a lift off roof is good. I got our hamster an extra large house (guinea pig size) so the flat top could double up as a shelf/level - when he had this house he built a huge nest inside it and also moved his toilet corner inside the house - so I moved his potty tray inside there too. So if they have a large house, or a house with 'rooms' or separate compartments in, they seem to instinctively treat it like a burrow - with areas for sleeping, toiletting and hoarding.
Labyrinth houses are also popular with hamsters, I think mainly because the central compartment is enclosed and dark. Our Syrian currently has a large house with a lift off roof and I put a tunnel over the entrance so it was dark inside.
Sometimes a hamster won't use a house that is provided but it's maybe because they have decided it's not big enough for a nest or not dark enough - they seem to like to be 'under' something to feel safe which might be why yours chose the wheel to nest under.
As Cypher says you can make a house out of a shoe box. If you want it to have a lift off lid you can use the lid of the shoe box and cut the base out so it is open underneath, and cut a door in one side. Putting a tube or tunnel over the door encourages them to use it.
Our hamster had a little plastic house originally that came with a shelf and although he seemed to like it, because it was round, it kept going smelly and mouldy inside with lack of ventilation and the top kept popping off because it wasn't big enough for him and his nest!
If it's for a Syrian, anything labelled 'hamster' is likely to be far too small and only suitable for a dwarf hamster. Rat sized accessories are best for Syrian hamsters.
This is our set up for our Syrian - it has a lot of big stuff in because he's a large syrian - for a dwarf hamster though it could look very similar except for a smaller wheel and smaller tube and the cage size is the minimum for any hamster - Syrian or dwarf.
This was his set up a year ago
And this is it now - it has stayed the same for quite a long time now as it seems to suit him - except I wish the cage was 100cm instead of 80cm so he has more floor space. The more space you have in the cage the better - it helps with enrichment so you can have areas for digging, a large house or nesting box and so on - people who can fit one in often have detolf conversions for hamster cages and these are over 5 foot long!
Here some examples of Ikea Detolf hacks - it's a glass cabinet on its side and you make a lid out of wood and mesh - it's a cheap option for a very large hamster habitat, and works out cheaper than most suitable cages.
https://www.google.co.uk/search?q=de...I0DRsQ_AUIBygC
What kind of hammy are you thinking of getting next?