Mounting the lens: Yes. Her Tamron will mount to a D5100 or D7000.
Metering: Yes. Her Tamron should provide metering with a D5100 or D7000. (Note: older and current AF lenses should have a chip, with electrical contacts in them, if the lens can meter with the N80, then she should be okay with either the D5100 or D7000.)
AF: It depends: D5100 does not have a built in screw drive motor in the camera body, which is what the Tamron needs to AF; it will only AF with AF-I or AF-S silent wave motors. The D7000 has the screw drive motor in the body and can AF with both the screw-drive system and the AF-S silent wave system.
Angle of View: The smaller DX sensor (24mm x 16mm) of a D5100 or D7000 will have a 1.5x crop on the images. A 50mm setting on that lens on her N80 will appear to be a 75mm lens on the D5100 or D7000. Great for telephoto when using FX (or 35mm film lenses) not so great at the wide end: at 28mm on a DX body will appear as a 42mm lens (similar to a "normal lens").
If she really like that focal range, then she should probably invest in an 18-200 DX lens.
DX lenses will vignette on film bodies (smaller image circle).