I am writing a novel and would like to know what hymn a family might have sung as their loved one died - in 1405 in England.
The singing of hymns as we understand them today wasn't necessarily a major part of medieval Christianity. As Ken Graham's answer says, the best known texts would have been those from the Requiem Mass, of which In Paradisum is probably the best known (along with Pie Iesu, the last part of the Dies Irae).
Another common text would be the Libera Me, sung both at the requiem and at the office of the dead.
These aren't really hymns as we conceive of them today, but texts set to chant.
However, you are talking about the family around a dying person. There is an abundantly likely text: the prayer Proficiscere Anima Christiana. It would most likely be said by a priest, but it is not inconceivable that a member of the family might have done so, especially in an educated family.
Here is the Latin text:
Proficiscere, anima christiana, de hoc mundo, in nomine Dei Patris omnipotentis, qui te creavit: in nomine Iesu Christi, Filii Dei vivi, qui pro te passus est: in nomine Spiritus Sancti, qui in te effusus est: in nomine gloriosae et sanctae Dei Genetricis Virginis Mariae: in nomine beati Ioseph, inclyti eiusdem Virginis Sponsi: in nomine Angelorum et Archangelorum: in nomine Thronorum et Dominationum: in nomine Principatuum et Potestatum: in nomine Virtutum, Cherubim et Seraphim: in nomine Patriarcharum et Prophetarum: in nomine sanctorum Apostolorum et Evangelistarum: in nomine sanctorum Martyrum et Confessorum: in nomine sanctorum Monachorum et Eremitarum: in nomine sanctarum Virginum, et omnium Sanctorum et Sanctarum Dei. Hodie sit in pace locus tuus, et habitatio tua in sancta Sion. Per eundem Christum Dominum nostrum
And my translation:
Go forth, Christian soul, from this world, in the name of God the Father almighty, who created thee; in the name of Jesus Christ, Son of the living God, who suffered for thee; in the name of the Holy Spirit, who was poured out in thee: in the name of the glorious and holy Mother of God the Virgin Mary; in the name of blessed Joseph, the glorious spouse of the Virgin; in the name of angels and archangels; in the name of Thrones and Dominations; in the name of Principalities and Powers; in the name of Virtues, Cherubim and Seraphim; in the name of Patriarchs and Prophets; in the name of the holy Apostles and Evangelists; in the name of the holy martyrs and confessors; in the name of holy monks and hermits; in the name of holy virgins, and of all the saints of God. Today may thy place be in peace and thy habitation in the holy Sion. Through the same Christ our Lord.
It would undoubtedly have been said in Latin, but I guess including it in translation wouldn't be a travesty.
Fourteenth to fifteenth centuries England was Catholic. The most common hymn sung by the faithful could possibly be the Latin hymn: "In paradisum". This hymn is still quite popular and I have sung it on many occasions as the body of a defunct has been lowered into the earth at a cemetery. Traditional religious still sing this at the graveside of their brethren.
Here it is played on YouTube: In Paradiusm - Traditional Catholic Requeim Mass
Another possibility could be the Sub tuum praesidium which has always been a popular devotional hymn since Christian antiquity. It is still a Catholic tradition to recite the De Profundis (Psalm 130) in Latin for those accustomed to the Latin to be recited when visiting the graveside of those we love.
Since the earliest times, Christians have sung "psalms and hymns and spiritual songs", both in private devotions and in corporate worship (Matthew 26:30; Mark 14:26; Acts 16:25; 1 Cor 14:26; Ephesians 5:19; Colossians 3:16; James 5:13; cf. Revelation 5:8–10; Revelation 14:1–5).
Non-scriptural hymns (i.e. not psalms or canticles) from the Early Church still sung today include 'Phos Hilaron', 'Sub tuum praesidium', and 'Te Deum'. - Hymn (Wikipedia)