The simplest way would be to put something enough heavy behind the laptop screen to make it lean on it.
lifehacks.stackexchange.com/users/21/nicael
you could make a hinge much like you find on a photo frame although this would have to be attached to the lid of the laptop and would look unsightly.
if that wouldn't be a problem then all you would need is a stiff piece of card cut into a triangle, and something to attach it eg.
duct tape if attached with a couple of millimetres between the lid and card it would fold to the side quite nicely
lifehacks.stackexchange.com/users/5015/samuel
I like large binder clips for problems like this.
Clip the black part to the edge of the lid, and adjust the silver "ears" so that they prop up the screen.
If the ears aren't long enough, use something like a thick book behind the laptop for the ears to lean on.
lifehacks.stackexchange.com/users/4939/GuitarPicker
Use some duct tape and some rubber bands.
If you use just the right size, it will apply just enough friction to prop the screen up.
If it will simply fall, glue or tape a small metal bar to the back, so it will stop falling if it reaches the case (the part with the CPU, etc).
lifehacks.stackexchange.com/users/82/QuyNguyen2013
Try setting it up against a wall.
That way, the screen won't tilt backwards.
lifehacks.stackexchange.com/users/93/Shevliaskovic
I had a similar problem which I initially tried fixing with black electrical tape.
However, the tape kept coming undone and the problem got worse.
Finally, in frustration, I used superglue.
To my surprise and delight, this fixed the problem!
lifehacks.stackexchange.com/users/11960/Pamela
Get a piece of styrofoam packing (which I suppose counts as ordinary household stuff, no?)
Carve it into an angled shape with a recess that holds the screen in a press fit.
lifehacks.stackexchange.com/users/1722/jamesqf
Does Lego count as a household item?
I'd build something from Lego to keep the screen in place.
With a few Technic beams and pins you can build a support that fits around the laptop, holding the screen at pretty much any angle you want, without risk of slipping away from the laptop and dropping the screen.
lifehacks.stackexchange.com/users/5307/Hobbes
Both of my hinges have broken.
I am about to order replacements.
For a temp fix, I have: black duct tape from the top left of the screen to the surface of my laptop's supporting table (or to anything that it will stick to, behind it), and tape from the top right of the screen to the front right of the base of the laptop.
These opposing tapes keep it fixed upright (again, this is temp), and they do not block the screen, although it is awkward getting to the numeric keypad on the right.
You will probably want to test the angle you need before affixing the two tape strips.
lifehacks.stackexchange.com/users/19250/Phil B
I have a laptop with a broken hinge. The broken hinge is cracked and exhibits no friction whatsoever when opening/closing the lid.  For now, the remaining hinge holds the screen in place. My concern is that this will stop working, allowing the screen to fall and crack.  Using  only  household items, how can I prop up the screen?
How do I prop up a laptop screen with a broken hinge?
lifehacks.stackexchange.com
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lifehacks.stackexchange.com/questions/83
lifehacks.stackexchange.com/users/15/Mooseman
The simplest way to hold the laptop screen in place is simply to place something heavy behind the screen and lean the screen against it. Alternatively, you could prop it against a wall, or use tape or lego to keep it upright. Lastly, you could even consider using superglue to glue it open. There are many solutions - you could put something behind the laptop to rest the screen on, make a hinge like on a photo frame, use large binder clips to hold it in place, use duct tape and runner bands, set it up against a wall, use superglue, cut a stand out of styrofoam, make a stand out of Lego or run strips of duct tape around it.
Put something heavy behind the laptop to lean the screen against. Make a hinge like you see on a photo frame. Use large binder clips - if necessary with a thick book to lean the ear on. Use duct tape and runner bands. Set it up against a wall. Use superglue. Carve some styrofoam packing into a triangular prism shape and lean screen on that. Make a support out of Lego. Use duct tape and stick it to the table and to the front of the laptop.