Looked at a lot of DIY bike trailers, and they seemed too unnecessarily complicated to build. I think you are better off starting with something existing and modifying it.
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Step 1: Start With Kids Bike Trailer
Start with this. I have seen these trailers from $20-$100 at garage sales. I had one that my kids haven't used in years. Time to repurpose this trailer.
Cut off the vinyl with a utility knife or scissors.
Step 2: Stripped
Now you have this.
Step 3: Construct the Trailer
Measure for a piece of 1/2" plywood. Notch out for the frame, because it fits better and you have more space this way.
Fit it on bottom frame. Then take it off and paint it.
Drill holes and secure plywood with nuts and bolts to bottom frame, using the existing holes on the frame.
Add eye hooks. Drill a small pilot hole for each eye hook, then screw them in.
I also put on two regular hooks in case I needed to loop a bungee cord around to make it tighter, which I have to do often.
Step 4: Back Side of Trailer
You can leave the upper frame around it if you want. I did because you could hang it on bicycle hooks in the garage easier. Also you can attach things on the frame. Plus people can see that you were smart enough to reuse an old kid trailer instead of spending more while completely building your own.
Step 5: Enjoy Your Trailer
I plan on using it for picnics, riding to the parade, going to the fireworks, kids baseball games, etc. Load up a cooler, blankets, etc. Use bungee cords to tie things down. Enjoy!
Good luck. Contact me with questions.