Unless I have misunderstood it is the eligible one that has applied. He is a full time student so cannot claim, she is a part time student so she can make a claim, thats probably why she has submitted the claim.
To make a claim she has to supply a NINO or make an application for a NINO and the DWP should give her one if she is applying for HB/CTB.
Once she gets over this hurdle you would need to consider her immigration status and I suspect she is likely to be subject to immigration control and not have recourse to public funds. You would then have to turn her claim down.
