Prayers for a deceased mother, like most other prayers for the dead, ultimately amount to asking God to grant the soul of the deceased eternal rest in heaven. Other sentiments and reasons may factor in, such as thanking God for the gift of the life of the departed.
Roman Catholics and Orthodox Catholics accept the practice of praying for the dead as founded on both Scripture and Apostolic Tradition, whereas most Protestants reject the practice as contrary to the Bible. The main issue lying behind this division is the Catholic doctrine of Purgatory, a state after death where souls who are worthy of heaven go and are purified by temporary suffering before entering eternal life.
The reason this has bearing on the question of how to say prayers, for example, for a deceased mother, is because there is really no reason to pray for the dead unless they can somehow benefit from the prayers of those on earth. Both Catholics and most traditional Protestants believe that heaven and hell are permanent states, and so if these are the only two states after death, as Protestants believe, then prayers for the dead are pointless. Catholics believe in Purgatory as an intermediate state, and believe that those who pray for the souls in Purgatory can ease their suffering. This is why prayers for the dead traditionally carry such notions as asking God to have mercy on the soul of the departed, or asking that they may "rest in peace."