There are generally two categories of this request: 1) asking for Mass to be said for some person or intention, and 2) asking for Mass to be said for a specific group of which you are a member -- that is, asking for the priest to celebrate a Mass at which your group is present. These two really are manifestations of the same desire: that a Mass is celebrated for the benefit of some person(s) or intention(s).
But what is the sought after benefit? When a group of, say, ten people asks a priest to celebrate Mass for them, what is that group after? Is the Mass an attempt to "build community" in the group? Is it a rallying point for them? Do they seek a private Mass for themselves so that it can be more personal and "close"? Is it a celebration of the group rather than of the Holy Sacrifice of the Mass?
Contrast that "benefit" (which is better described as "edification") with this: that your group is on the priest's mind -- you are his "intention" during the offering of the Mass -- when the Holy Sacrifice of the Mass is made. In the right rendering of worship to God the Father through the perpetual sacrifice of His Only-Begotten Son, our Lord Jesus Christ, the priest will have your group in mind.
Let me share a lengthy quote from an essay written by Shawn Tribe of The New Liturgical Movement, entitled "The Nature and Social Implications of the Liturgical Act" (emphasis mine):
The Mass is not first and foremost a tool for Eucharistic piety or adoration. Eucharistic piety is a noble thing of course, and one of the most solemn moments of the Mass finds us adoring the Body and Blood of Christ, but that is not the primary end of the liturgical act. Neither is the Mass primarily a vehicle for the reception of the Eucharist -- though frequent and worthy reception of the Eucharist brings with it many important graces of course. The Mass is not even primarily about our own sanctification. To comprehend the essential nature of the Christian liturgy bears minding the nature of the Jewish Temple liturgies and their sacrificial offerings and how that imperfectly foreshadowed the Christian liturgy and Sacrifice of Christ. The Christian liturgy is first and foremost an act of rendering due worship to God the Father through the perpetuated sacrifice and offering of God the Son. It is this sacrificial nature of the liturgy and the worship of God the Father that particularly drives the liturgical act. It is important that we have this proper understanding of the primary end of the Mass for all else flows from this. This is also why we should not make Eucharistic piety to be the be-all and end-all of the liturgy. In point of fact, worship and sacrifice are the be-all and end-all of the liturgy.Now, at first glance, this seems strange to us. The primary end of Mass isn't our reception of the Eucharist? It isn't for Eucharistic worship? It isn't even about our own sanctification? Yes, as odd as it sounds, even if there was nothing for us to "get" from the Mass, it would still be dignum et iustus ("right and just") to offer the Holy Sacrifice of the Mass to God the Father. Mass is not for our edification (although we can certainly learn from it and be sanctified by it), it is for God's glory.
Therefore, a Mass said for a group is said on behalf of that group particularly: the group and its members are the intention of the Mass. The primary end of the Mass is the rendering of worship to the Father through the sacrifice of the Son, and that sacrifice is being re-presented with the intentions of your group in the mind of the priest. Our sanctification, though not the primary end of Mass, is bound tightly to the primary end of Mass, worship of God the Father.
And so I return to the question at hand: why do we ask for a priest to "say Mass for us"? Is it for our edification -- a "personal" or "special" Mass -- or is it for the glorifcation of God in the context of our need for the sacrifice of His Son?