I've done numerous amounts of paper 3d models, and I use 150gsm (grams per square metre) paper, that holds it's shape rather well. I don't know if you want the light to shine through the paper, or be reflected off it downwards or what ever, but the thicker the paper the less the light shines through.
The other two answers (so far) have the correct information, that I've learnt while playing around with paper modelling, craft and lamps so read them for more information.
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If you're following this https://www.instructables.com/id/LED_Paper_Craft_Lamps/ to the letter: I would suggest a matte white cardstock if you have an inkjet printer. It will absorb the ink better than glossy and make it easier to work with.If you're working with a color laser printer or having it printed at a copy shop, go for glossy white cardstock.Regular weight paper will work too, but wont be as strong. Photo paper would be a happy medium, but I can't recomend it as it tends to be soft and fold weird.The key is that you need a stiff paper. Card stock is also called cover stock, heavy weight, etc.. You can buy it anywhere that sells office supplies.
To begin with: low power LEDs at most get 'warm' - make sure to build your LED array powered such that it doesn't ever get too hot. Use proper heat-sinking where possible/appropriate.
Paper-wise, there are plenty of choices in craft stores, many will be coated with a fire-retardant material. This should be noted on the packaging. Depending on the mood of your light - translucent or opaque - you'll choose whatever paper you want.