Her Christian family are organizing a running contest in memory of their deceased loved one; what should she do about that?
If a person dies following a religion of kufr, whether he was a Christian or a Jew, then he is one of the people of hell and will abide therein for eternity. Allah, may He be glorified and exalted, says (interpretation of the meaning):
“Verily, Allah forgives not that partners should be set up with him in worship, but He forgives except that (anything else) to whom He pleases, and whoever sets up partners with Allah in worship, he has indeed invented a tremendous sin”
“Surely, they have disbelieved who say: "Allah is the Messiah (‘Eesa (Jesus)), son of Maryam (Mary)." But the Messiah (‘Eesa (Jesus)) said: "O Children of Israel! Worship Allah, my Lord and your Lord." Verily, whosoever sets up partners in worship with Allah, then Allah has forbidden Paradise for him, and the Fire will be his abode. And for the Zalimoon (polytheists and wrongdoers) there are no helpers.
Surely, disbelievers are those who said: "Allah is the third of the three (in a Trinity)." But there is no ilaah (god) (none who has the right to be worshipped) but One Ilaah (God -Allah). And if they cease not from what they say, verily, a painful torment will befall the disbelievers among them.
Will they not repent to Allah and ask His Forgiveness? For Allah is Oft-Forgiving, Most Merciful.
The Messiah (‘Eesa (Jesus)), son of Maryam (Mary), was no more than a Messenger; many were the Messengers that passed away before him. His mother (Maryam (Mary)) was a Siddiqah (i.e. she believed in the words of Allah and His Books (see Verse 66:12)). They both used to eat food (as any other human being, while Allah does not eat). Look how We make the Ayat (proofs, evidences, verses, lessons, signs, revelations, etc.) clear to them, yet look how they are deluded away (from the truth).
Say (O Muhammad SAW to mankind): "How do you worship besides Allah something which has no power either to harm or to benefit you? But it is Allah Who is the All-Hearer, All-Knower”
“This is the Guidance of Allah with which He guides whomsoever He will of His slaves. But if they had joined in worship others with Allah, all that they used to do would have been of no benefit to them”
In Saheeh Muslim (153) it is narrated from Abu Hurayrah (may Allah be pleased with him) from the Messenger of Allah (blessings and peace of Allah be upon him) that he said: “By the One in Whose hand is the soul of Muhammad, no one among this nation, Jew or Christian, hears of me then dies not believing in that with which I was sent, but he will be one of the people of the Fire.”
As that is the case, if a person dies in a state of kufr, his good deeds are rendered invalid before his Lord and none of them will benefit him before Allah in the hereafter; similarly, no action, intercession or supplication on the part of other people will benefit him.
Allah, may He be exalted, says concerning the situation of the people of Hell (interpretation of the meaning):
“And it will be said to them: "Where are those (the false gods whom you used to set up as rivals with Allah) that you used to worship
Instead of Allah? Can they help you or (even) help themselves?
Then they will be thrown on their faces into the (Fire), They and the Ghawoon (devils, and those who were in error).
And the whole hosts of Iblees (Satan) together.
They will say while contending therein,
By Allah, we were truly in a manifest error,
When We held you (false gods) as equals (in worship) with the Lord of the Alameen (mankind, jinns and all that exists);
And none has brought us into error except the Mujrimoon (Iblees (Satan) and those of human beings who commit crimes, murderers, polytheists, oppressors, etc.).
Now we have no intercessors,
Nor a close friend (to help us).
(Alas!) If we only had a chance to return (to the world), we shall truly be among the believers!
Verily! In this is indeed a sign, yet most of them are not believers”
Honouring the memory of the deceased by organising a marathon is not part of the way prescribed by Allah; rather it is something that has been introduced by non-Muslims according to their way of honouring the memory of the dead. It is a false kind of honour for which there is no basis in the religion of Allah. Even if there was some basis for it, we have stated above that if a person dies in a state of disbelief, no deed will benefit him in the hereafter; the deceased will not benefit because of his disbelief and that of those who do this action. This is darkness upon darkness.
In Islam there is no kind of honouring the dead or celebrating their memory; rather what Islam prescribes is to remember the deceased by praising him, praying for forgiveness for him, offering supplication for him, giving charity on his behalf, and other kinds of good deeds the reward for which will reach him, if he died in Islam.
According to Islamic rulings, competitions fall into three categories.
Ibn ‘Uthaymeen (may Allah have mercy on him) said:
1. Those that are haraam, with or without prizes, such as competing in something that is haraam.
2. Those that are halaal if no prize is offered and are haraam if a prize is offered, such as foot races and the like, in which it is permissible to compete.
3. Those that are halaal with or without prizes, such as competitions in archery and horse and camel racing.
End quote from Fataawa Noor ‘ala ad-Darb, 9/311
To sum up:
It is not prescribed to take part in activities to honour the memory of the dead among the disbelievers, or to keep their memory alive. If you cannot prevent that, then at least do not take part in that and try to excuse yourself from your family whilst avoiding confrontation with them. If they respect your religious feelings, then this is good, otherwise give them any excuse that they will accept from you.
And Allah knows best.