Most parishes, by the way, do have very nice cards informing the person for whom the Mass is being offered of the fact, as well as the name of the person having the Mass offered. They are the size of a smallish greeting card. Depending on the individual parish, they may mail them to the recipient, or you can do that yourself, so it’s a good idea to have the person’s address with you. At many churches you can ask for a Mass to be celebrated at a particular day and time when you have the Mass said for someone.
Some shrines and religious congregations or orders have something called a perpetual Mass enrollment. For a small offering (but more than one Mass stipend,) you can have a person enrolled in such a society. The recipient gets a nice card in the mail, frequently beautifully decorated, that informs them that they will be remembered in Masses said by a priest in that order at a specified interval for a set period of time. Some of these societies will keep the person on the Mass request list permanently (hence, perpetual Mass.) There are even a few such societies where you can offer for a specific number of people, and they will send you a box of cards. When you use one, you contact the society and get a replacement one. You can ask at your own parish, or google “perpetual mass enrollment” to find the websites of such congregations.
A Mass said for any person, living or dead, is the highest form of prayer we can offer. The advantage of having one offered for a living person is that that person, being alive, is capable of responding to the graces to be garnered by that particular Mass being said for them.