What would you say is going to be acceptable backlash when engraving letters that are approximately 1/8" tall? I'm building a small machine for doing pet tags for a non-profit animal rescue group to raise money.

I would say that none would be best and it goes downhill from there. Those are really small letters. Any major backlash will surely show without compensation. Mach 3 can compensate for backlash to help you achieve acceptable results. The more backlash you have though, the less effective it may be. I think any ball screw setup will work well. The typical rolled ball screws may have about .004" accuracy over 1 foot. I think that is plenty good for lettering and engraving, especially with comp.
Acme screws with anti-backlash nuts may also be a good choice. The Dumpster nuts look good to me and most like them. Cheaper than typical ball screws. Fine threaded rod can even be made to work fairly accurate if they have anti-backlash nuts and you have the right motors driving them.

You need to take into account the rest of the machine too, though. Linear bearings are the best I think. Then there are other methods that can be made to work well, like skate bearings, v bearings etc.

Steel or cast iron might give the best rigidity, but aluminum, plastic and even wood and wood products can be used. Depending on design, these will all have varying accuracies and flex.

It is pretty easy to build a small accurate mill or engraver. Look at some of the project logs to help you determine what type you want if you are wanting to build your own. Most of the completed projects have the results that were achieved so far. Some are surprisingly good.