The interesting question about mourning is not what is done on the day but for how long afterwards, and traditions answer it with remarkable precision.
Jewish mourning is a descending staircase. Three days of the most intense grief; seven of shiva, sitting at home while the community comes; thirty of sheloshim, with restrictions on celebration; and for a parent, eleven months of saying the mourner's prayer daily, which requires a quorum and therefore requires the mourner to be among people every day for the better part of a year. Thereafter the anniversary is kept with a lamp lit for a full day. The structure is often described as psychologically shrewd by people who do not otherwise find Jewish law shrewd, and the observation is fair: it prescribes withdrawal, then re-entry, then permanent low-level remembrance, and it does not leave the timing to the bereaved.
Hindu shraddha offerings run over twelve days and then annually, and they are framed as an obligation the living owe rather than a comfort they receive.
Orthodox Christians keep the third, ninth and fortieth days and the anniversary, with boiled wheat sweetened and blessed, the grain standing for the seed that is buried before it rises.
Chinese and Japanese Buddhist households observe seven memorials at seven-day intervals, on the understanding that the intermediate state lasts forty-nine days and that what is done during it matters. Obon in midsummer brings the ancestors home for a few nights, with lanterns to guide them in and out.
The Catholic month's mind and the commemoration of All Souls do comparable work in a tradition that also prays for the dead year-round.
One observation from survey data rather than from theology. Repeating memorial observances are frequently the last element of a religious tradition that a secularising family retains, and they show up in the numbers long after belief in an afterlife has fallen away in the same population. The secular memorial service, now common in Britain and North America, is the endpoint of that: it keeps the gathering, the speeches and the timing, and quietly drops the claim that anything is being done for the person who died.