No offense guys, but some of you just aren't hearing the part about Grandma spending limit. You need to remember she's not as young as most of you guys, and not a total bicycle geek (not yet, anyway). I think I know where she's coming from; I'm 50, and my sister-in-law and her husband are in their late 60s. They bought a matching pair of Raleigh Sports back in the early '70s, rode them for a few years, then as babies started arriving they relegated the Raleighs to the back yard for the next quarter-century.
Now the kids are grown, sis got a pair of new titanium hips last year, and they decided they'd like to try riding again. They asked me if I could return the Raleighs to a useable condition, so I took them home, I pulled them apart, repacked and adjusted all the bearings, threw on a set of cheapo WalMart tires, did as much cosmetic work as elbow grease would allow, and one bike needed a saddle. Now they're both in good, safe running order and I spent less than $75 on both of them. They don't look half bad either, for all they've been through.
Now, there are a dozen good reasons why my old coots would be better off with a pair of new comfort bikes; you guys all know them -- less weight, better brakes and shifters, suspension, etcetera. But here's the thing: There's a good possibility they'll ride the bikes a few more times and decide they're either not up to it physically, or it's just not as much fun as they remember. If that's the case, they aren't out a lot of money and they got their Raleighs (which have a certain sentimental value to them) redone. If by next summer I see they're riding regularly, then I'll drag them down to the LBS and let them get acquainted with the 21st Century (if they haven't gone there on their own already). Right now though, it'd be dumb for them to lay out $600 -$700 for a pair of new bikes.
I'm guessing the case with Redhed's grandmother is really similar. My advice is to put her old bike in safe running order with a minimal investment. If that isn't possible, you can get her a decent thrift store bike for $40 or less that will let her find out how serious she is about riding. If it turns out she sees that riding a bicycle is just like 'riding a bicycle', take her to the LBS and get her a ride on a modern bike. I'll bet that $100 limit will fly right out the window.