an she contact the court before the hearing to plea for an out of court settlement (and thus avoid the hearing)?
The court doesn't handle out of court settlements (clue is in the name

) she would need to speak to the prosecuting authority (seem to be the TOC) and see if they'll agree to drop the action.
[Usual proviso that I'm not a lawyer, this isn't legal advice and I'm just a bloke on the internet]
@9mcbrides I think it's perhaps worth clarifying something (and let me preface this by saying I do not support, at all, the present regime when it comes to fare evasion). There are two main piece of law that are used when it comes to fare evasion on the railway. There is the
Byelaw 18 of the Railway Byelaws (what your daughter has been charged under) and there is
Section 5(3) of the Regulation of Railways Act 1889 (that's not a typo the law really is from 1889!).
Whilst fundamentally both deal with fare evasion there is a fairly fundamental difference between them. Byelaw 18(1) requires that "no person shall enter any train for the purpose of travelling on the railway unless he has with him a valid ticket entitling him to travel" and 18(2) requires that you show that ticket on demand. That's basically it. If you cannot show a ticket on demand you have committed an offence (unless one of the limited exemptions in 18(3) apply). It is for all intents and purposes a strict liability and binary choice. Were able to show a ticket entitling you to travel? Yes, no offence committed. No, offence committed.
RoRA 1889 S5(3) is a slightly different kettle of fish however. It requires that intent avoid the fare be shown "If any person travels or attempts to travel on a railway without having previously paid his fare, and with intent to avoid payment thereof". This is higher bar for the prosecuting authority to jump over than the simple "Yes/No" question that Byelaw 18 poses. And one which you daughter would likely have been able to mount a reasonable attempt of defending herself against with the available evidence (nothing is certain however).
So effectively by choosing the Byelaw offence rather than RoRA they are conceding that your daughter probably wasn't intentionally evading her fare and that they certainly couldn't prove that she was.
My personal view is that Byelaw 18 is a terrible rule and should be ditched immediately (indeed the whole way we deal with fare evasion should be subject to a fundamental overhaul). However the world is not what we would like it to be in this case. My view is that your daughter is, no matter how unjust it is, guilty of breaching Byelaw 18 and if she goes to court she will most likely lose (unless she pays for a very good solicitor to represent her).
My recommendation, therefore, would be to contact the prosecuting authority and apologise for what's happened, that your daughter will take care to ensure that she doesn't misplace her ticket again in the future and will be willing to pay for any reasonable admin costs incurred and for the outstanding fare. Which, without swearing, is rubbish. But I don't personally see another way around it as I have very little confidence that showing up at court and presenting the evidence that the fare was paid will actually be sufficient to prevent a conviction due to the appalling way that the Byelaw is drafted. Meanwhile paying for a solicitor who might be able to get her out of this predicament I would bet will end up being more expensive in the long run.
I would also suggest she write to her MP to tell them what's happened to her. It is the Government through the DfT which sets the laws and Byelaws the railway uses so they are the ones with the power to change it...