Since you are in RCIA, you are no doubt curious and interested to learn about Catholic worship and practices. I wish all Catholics were interested for their whole lives in learning about our worship.
In 2008, the Congregation for Divine Worship and the Discipline of the Sacraments received a letter asking precisely this question. The congregation responded in a private reply with five observations on why this practice is not permitted. [NOT]
But first, let me note that even though private replies do not have the force of universal law, they typically (and this one especially) contain an excellent analysis and resolution of the issue, giving us a unique look at the practice of the Roman Curia. In other words, this private reply is persuasive not by reason of authority but by the authority of right reason, to which every well-intentioned Catholic should submit. Here are their five observations:
Blessing given at end of Mass
[1] The Congregation for Divine Worship points out in their first observation that the liturgical blessing of the Mass is given to everyone gathered in the church just a few moments after the distribution of Holy Communion. This occurs when the priest, making the sign of the cross, says, “May Almighty God bless you, the Father, and the Son, and the Holy Spirit.”
In other words, there is no need to bless only some members of the congregation (e.g. children and non-Catholics) during communion, when the entire congregation is blessed by the priest just moments later.
Laity unable to bless at Mass
[2] In the second observation, we are reminded that within the context of Mass, blessings are the competency of the priest, not lay persons. Article 18 of the Book of Blessings [ugh] notes that even though lay persons may give some blessings, “whenever a priest or deacon is present, the office of presiding [over a blessing] should be left to him.” [And at Mass there is always a sacerdos present.]
A 1997 instruction, Ecclesia de Mysterio, on the collaboration of the lay faithful further indicates that the laity should never say prayers or perform actions during the Mass which are proper to the priest, as this may lead to a confusion of roles. Since the blessing of the congregation during Mass is reserved to the priest, lay persons must avoid doing so.
Laying on of hands discouraged
[3] The third observation addresses the practice in some places where the EMHC lays hands on a member of the congregation as a sign of blessing. The private reply states that this practice “is to be explicitly discouraged” because the laying on of hands has its own “sacramental significance” which is inappropriate here. The Catechism notes that since this specific sign commonly accompanies the administration of sacraments (e.g. Confirmation) and the succession of the apostles, the laying on of hands must not be used here.
Some prohibited from receiving blessings
[4] Finally, in the fourth and fifth observations, the private reply notes there are some who should neither approach Holy Communion nor receive a blessing. This would include non-Catholics and those mentioned in canon 915 of the Code of Canon Law, such as those under the penalty of excommunication and those persisting in manifest grave sin. [!] Giving a blessing to these persons might give the impression that they are in full communion with the Church or have returned to good standing. In order to avoid the possibility of scandal, EMHCs should not give blessings.
Additions to the rite prohibited
[5] Finally, even though the private reply does not specifically mention this, we ought to recall that “no one may on a personal initiative add to or omit or alter anything in [liturgical] books” as canon 846 of the Code of Canon Law clearly states. Nowhere in the Roman Missal or the GIRM are the extraordinary ministers of Holy Communion instructed to bless those unable to receive communion; therefore, this practice of blessing is one of these additions to the rite which is strictly prohibited.
Making use of the sacramentals
Sometimes we may be tempted to think that since something is not part of the Mass it has no spiritual importance. But this would be to neglect the power of the sacramentals, such as blessings, which are liturgical actions signifying spiritual effects obtained through the intercession of the Church. Done properly and in the right context, these blessings better dispose us to receive grace and sanctify various occasions in life.
[NB] One such sacramental that lay persons may administer is the blessing of sons/daughters, which can be as simple as praying over your children: “May the Lord keep you and make you grow in his love, so that you may live worthy of the calling he has given you, now and for ever. Amen.”
Therefore, even if EMHCs are not permitted to give blessings during Mass, the desire to bless is good nonetheless and can become a fruitful aspect of our faith when done in accordance with the Church’s rites. As a parent, I have always enjoyed the practice of blessing my young children before bed and teaching them to reverence the Eucharist with a simple bow of the head as they walk past the minister of Holy Communion at Mass.