Can one take up the cross of someone who is suffering from illnesses and carry them like Simon of Cyrene and pray that to God? If family members are sick and one wants to pray for their recovery and carry their crosses can they pray to God to bestow their suffering on him or her?

Is that a valid prayer? 

Can one take up their crosses instead like Simon of Cyrene?
Romans 15:1:


  Now, we that are stronger ought to bear the infirmities of the weak and not to please ourselves.


Galatians 6:2:


  Bear ye one another's burdens: and so you shall fulfil the law of Christ.


Fr. Andrew Pradel, O.P., St. Vincent Ferrer: Angel of the Judgment ch. 11 ¶15:


  [Saints] are not content with sharing the sufferings of their brethren, but joyfully take the whole on themselves. Thus the Seraphic Virgin of Siena [St. Catherine of Siena] took entirely on herself the pains of Purgatory which her father ought to have suffered; thus also did St. Michael-des-Saints of the Order of the Trinity, with God’s permission, undertake a malignant fever from which a friend of his was suffering. The Great Model of Saints gave the first example of this. “Surely He hath borne our infirmities,” says the prophet, “and carried our sorrows.” (Isaias 53:4).


Also, suffrages are when we offer our Masses, prayers, sacrifices, or acts of piety for the deliverance of the souls in purgatory.