The Nikon D3300 takes lenses with a Nikon F mount. I would also recommend looking for lenses that have focus motors built in (AF-S on Nikon brand lenses, USD on Tamron or HSM on Sigma), as the D3300 does not have a focus motor in the camera body. You can use lenses without focus motors but on the D3300 they would be manual focus only.
In the ranges your cousin has asked about, you might consider:
Tamron AF 70-300mm f/4.0-5.6 SP Di VC USD XLD
Nikon AF-S 70-300mm f/4.5-5.6 G IF ED VR
Nikon AF-S 55-300mm f/4.5-5.6 G ED VR II DX
Nikon AF-S 18-300mm f/3.5-6.3 G ED DX VR II
Please note, I am not making any recommendation as to which, if any, of these lenses would be most suitable, as I have no idea what your cousin would like to use the lens for.
You could also try asking your cousin if he has any preference, as he might have had a specific lens in mind and simply assumed you would know which one he wanted.