Don't confuse 'zoom' with 'Telephoto'.
The focal length of a lens is usually quoted in mm. On a 'crop-sensor' DSLR 35mm is the 'Normal Angle' lens length, ie neither wide angle nor telephoto.
- Anything shorter, say a 15mm lens would be 'wide angle'.
- Anything longer, say a 100mm lens would be a 'telephoto'.
A 'Zoom' lens is merely one that has a variable focal length... it can be either wide-angle OR telephoto, and frequently is both... eg the standard 18-55mm variable focal length 'zoom' lens that normally comes with entry level DSLR's in the 'kit'... so she likely already has a 'zoom' lens, and one that is both a 'wide angle' at the 18mm setting, a "telephoto" at the 55mm setting and a 'normal angle' at the 35mm setting....
Oh-Kay.... I say get this clear cos it IS sort of a bit fundamental.
I have a bag load of 'prime' lenses, that is to say they are 'non' zoom lenses; they each have a fixed focal length, and most belong to one of my old film cameras; But, the bag has lenses from 12mm focal length up to 300mm focal length; they don't 'zoom', but the longer ones, particular on an adapter on the DSLR are 'Telephoto'.
I suspect... what she wants is a longer 'Telephoto' lens, not necessarily a 'zoom' lens, which as said, she likely already had in the 18-55mm 'Kit' lens that likely came with the camera.
So how 'long' does she need/want in the telephoto range?
The Nikon 'kits' at the budget end used to often contain two lenses; the standard 18-55mm 'normal' zoom, and either the 55-200, telephoto zoom, or the later 55-300mm telephoto zoom. Both lenses, new are sub £200 at last look, the 55-200 often available for around £100 or so, new, and both cheaper still 2nd hand.
Of the two, the 55-200 is a very good VFM lens, the 55-300 offers that much more 'zoom' or 'reach' and is more expensive for it; but both still pretty cheap lenses, and genuine Nikon/Nikkor. Either would probably be good enough and do the job.. and if she really needs bigger, better, faster, more... then that's an upgrade for next time about... either lens can be sold on for no great loss, especially of bought 2nd hand to start with.
70-200 or 70-300, is a zoom range more suited to 'Full-Frame' sensor or film cameras. As such, the disparity in prices can be even greater, because the lenses for older film cameras will be manual focus only and likely 2nd hand and 20+ years old to start with, or they will be brand new all electric auto-focus lenses intended for modern, professional grade, Auto-Focus, 'Full-Frame' DSLR's.
Here Nikon have, for 60 odd years, made a sales pitch based on maintaining the compatibility of their 'F-Mount' lenses, so there are a HECK of a lot of lenses out there that could be fitted to this camera, good, bad and ugly, from Nikon themselves and any of the 3rd party independents like Sigma or Tameron or many others... how good they may or may not be probably varies as much as price and the two are only notionally related!
So, to return to the Nikon AF-S DX Zoom-Nikkor 55-200mm f/4-5.6G ED... 55-200 is the zoom-range of focal lengths. AF-S means its an auto-focus lens with in-lens focus motor the D3500 needs as it doesn't have an in-body fous motor. f/4-5.6 is the maximum aperture; It has two numbers, because the f-no is a ratio of the hole, (Aperture!) diameter to the focal length; so if the focal length changes with 'Zoom' so does the ratio to the hole diameter; so f4 is the max aperture at the short, 55mm focal length setting, f5.6 is the max aperture at the long/telephoto, 200mm focal length setting.
The Nikon AF-S DX NIKKOR 55-300mm f/4.5-5.6G VR, pretty ,much the same deal, only it has an extra 50% more 'zoom'. The VR, by the way, stands for 'Vibration Reduction' and both come in either VR or imroved VRII versions.. at extra cost, IF you deem VR essential or worth-while.. personally I don't, much, but still,
The 55-200, new is priced around £100 or so; the 55-300, new around £200, so twice the price, which is why I say that the 55-200 is cracking VFM.
Personally I have the 55-300.. and that extra 100mm of 'zoom' REALLY don't get a lot of use, and isn't worth the extra to my mind. Its also not a great lens, but it engineered to an optimal 'acceptable quality level' and for a beginner on a beginners camera, that is probably more than good enough. The 55-200 is likewise not a great lens, but it is about as good, and for half the money!
As mentioned I have a bag of primes for old film cameras I can fit to the Electric-Picture-Maker via an adapter. Taking back-to back comparison shots with some of these has shown just how much the image quality of the Nikkor 'Kit' lenses has been sacrificed for cost and coverage... but unless you have £1000's to spend on alternative lenses and do back-to back comparisons.. you would never know!!! and them kit lenses, as said are more than good enough to be getting on with!
SO, my recommends would have to be the 55-200, for around £100, its a lot of 'telephoto' zoom... to suck it and see, and IF it aint enough, either a fast enough f-number, or a long enough focal length or whatever.... well, its a start, chop it in for something bigger better faster more when better clued up as to what more is more important.
FWIW, my daughter is now at Uni studying Photography; she started out with a Nikon D3100, and an AF-S 35mm 'prime' lens. She had, for GCSE and A-Level photo, little cause to use a telephoto, at all... and still doesn't, but when she did, she rarely needed more than 150mm, and frequently, for the magnification and IQ, often stole my old 135mm 'prime' for film camera... so don't be hung up on the idea you have to get as much 'zoom' as you can; as said, that 55-200 will likely be more than up to the task for a long while, and it be cheap!
