What is the best base/foundation for a shed to discourage rodents ?

I'd go with a poured cement foundation. Just my personal preference. Best of luck.

Pour a concrete pad is the best. If not possible, concrete pavers work well.

Hardy backer, board, like you would use behind or under tile

The foundation will do just so much. Mice and rats will chew their way into a shed. Every crevice and small hole needs to be caulked. You really need to keep it as clean as possible with everything they might like to eat put in metal cans. Rats can chew through plastic tubs. If there are no pets that can get to it, use a rodent bait. Rat Terrier dogs are good to have as outside dogs to help keep rodent population in check.

CRUSHED HOT RED PEPPERS MY GRANDFATHER HAD MANY GARDENS NO DOGS OR RODENTS KEPT THEM AWAY FROM DOING THEIR THING??

A cement poured base and put the bolts in the cement to attach shed