Let your child decide what to cut up and make a puzzle from - it could be a cereal box, leaflet or other piece of card from the recycling. Ask them about why they have chosen that picture to use.
At least if your family likes jigsaw puzzles. The idea is very simple, laminate a photograph onto some laser-cuttable board, and then run the laser over the outline of the pieces. Bam! Instant puzzle.
Cut out the puzzle pieces and stick them to a piece of cardboard to create your own dinosaur jigsaw. Muddle them up and see if you can put them back together.