It is accepted that European vehicles built in the mid-to-late 90s were superior safety-wise to the Asian brands rolling off the assembly lines at the same time.
Reliability was a different story, however. Japanese carmakers especially produced cars which required fewer visits to the garage than the Euro brands - though it never pays to assume every Japanese car is bullet-proof.
An older vehicle doesn't have to be an unsafe car, provided the driver is taught to understand some common road-handling skills. Reading the road, understanding technical jargon such as over-steer and under-steer, learning how and when to brake and corner will improve driver skills and provide some peace of mind.
There are companies offering advanced training at affordable prices (www.nationaladvanced.co.nz is one I would recommend).
For those on limited budgets the safest vehicle can often be the one with the better-educated driver.
I suggest you stick to the tried and proven Japanese models. Avoid Constant Variable Transmissions (CVT) in this price bracket and if buying privately look for long-term ownership with a proven service history. Buying from a reputable licensed trader (dealer) should not be ruled out. You may pay a little more but you may gain protection from the Consumer Guarantees Act.
The budget: $5000
2000 Toyota Echo/Vitz
Essentially the same car; the NZ new versions were called Echo and the used imports Vitz. Don't drop below 1.3-litre and if she can handle a manual all the better for power, economy and long-term reliability.
2000 Hyundai Accent
Very few used imports available so a good chance of finding one with long-term ownership, reasonable mileage and proven service history.
2000 Nissan Pulsar
Proven reliability and good value for money. Check thoroughly for body corrosion.
Driven recommends
In this price bracket invest in Advanced Driver Education training, regardless of make and model.