What type of wood can be used for a jigsaw puzzle?

If you want to go with a solid wood then a close-grained or tight-grained (these mean the same thing) hardwood would be ideal. There are numerous suitable species you can select from.

Hardwood
Judging by your use of Imperial units in the Question I'm guessing you're American so maple is likely to be available to you at a reasonable price and it's a domestic hardwood, so more green/sustainable than many alternatives if that appeals to you¹.

Maple comes in two broad types, hard maple and soft maple. Both are hard hardwoods. As you'd expect hard maple is harder, but soft maple is more than tough enough for this application, in addition to often being cheaper. So if you have a choice I'd go with the latter. In general it'll be noticeably easier to cut than hard maple, which may be a significant factor depending on how you will be cutting your puzzle pieces.

Other options include birch, poplar (a little of the soft side), and cherry.

Ply
I say IF using solid wood to begin with because plywood is an option. From what I've seen many large-piece modern wooden jigsaws are made from a good type and grade of hardwood ply. Birch plywood is often the best available in any given location, with the tightest construction and fewest internal flaws (perhaps none).

As I heard of it only yesterday on Ron Paulk's YouTube channel I'll mention a new plywood, called Premcore®Plus. It's available with either maple or birch veneers, in a wide range of thicknesses. Paulk says it is the best plywood he's ever used, while being cheaper than many competitor products!

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¹ Far fewer carbon miles, no worries about depleting a badly-managed tropical source of hardwoods.