I have a '86 Samurai that is slightly lifted.

I tow it four down on the hwy.

Off road in the Mojave desert.

I'm looking for suggestions on tires that have puncture resistant sidewalls.

I have already had three side wall punctures from desert brush in the tires that were on the Zuki when I bought it. They are AT but more on road than off.

Thanks!

I'd say don't scrimp on tires. Go top of the line BFG, Michelin, or Goodyear in the M/T or A/T and you should be OK.

Good luck, Eric

What tires are you running now - are they LT (light truck) or P-Metric (passenger car) - AT tread are available in both?  If you're currently running P-metric, then any LT tire will have stronger sidewalls.  If you're already running LTs then you may need to look at the load rating, but things can become tricky, the higher the load rating, the stronger the sidewall, but, that strength comes at a price, increased cost, increased weight and increased stiffness.

Look at the hankook ATs.  They come in a 4 ply and a 10 ply, take you pick.  The ten ply would be harsh on the road for such a light rig, but they would help you keep the sticks out.  I’ve been running Ironman mud tires on my big truck and like them a lot for the price. 
BFG ain’t what they used to be and I’d stay far away from those at any price. 

Thank you all for the suggestions.

I have no problem paying for good tires. I go quite remote so the more dependable the better.

Currently it has GY LT.

I have no problem with stiff side walls. The Zuki already rides like a tank.

Definitely hankook all the way.  They are a great value and a great tire.  Read the reviews.  They look great and wear awesome. 
For mud tyre Toyo makes the best I’d say, but honestly there are many.  I love a swamper as always, but the hankook mud tyre is also superb.  My Suzuki gained 4mpg going from swamper to hankook muds

The Hankooks look like a great option.

Thanks!

I have been running Cooper STT  Pro 31/10.50/15 on my Samurai for 2 years in mud snow sand and Eastern Oregon desert two track and have had absolutely nothing but good times. I love em.