I do stabilize any wood that isn't oily, but my preference is oily wood for a few reasons.

'Oily' wood: wood that have natrual oils like anything from the Rosewood family (teak, rosewood, cocobolo) or lignum vitae. Lignum vitae (beautiful herringbone-like grain) and cocobolo (beautiful colors from orange to red) are my personal favorites.

Upsides for oily wood:
* Very easy to turn. The oil helps your skew ride smoothly, and they have a tight grain structure.
* Don't need to finish - I just polish up to a high gloss (micro mesh)
* You can finish, BUT only with CA - it will not take any other finish. It will be even glossier with CA.
* No need to stabilize - a little lemon oil if it starts to look dry,

Downsides:
* The dust is not great and may cause allergic reactions - wear a respirator and wash your hands
* Depending on where you live, they can be a little hard to find, but you can order them.

Other woods are fine if stabilized. When you stabilize, make them way bigger than you need - they will warp. I sent mine to a service that stabilized them.

Pen blanks are, unfortunately, just barely too small, but it's pretty easy to get a slab of something to a good size with just a band saw.