Yes. You can announce your intentions, verbally or mentally, before you begin a rosary.
We can offer up anything and everything for any special intention:
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postive:
Good works, prayers, etc., you can even “offer” a mass yourself, privately, at a mass you attend (and traditionally have the priest offer for your intentions for a mass you will not attend–remember the mass card?!)
Good info here (LINK), especially: " . . . the Church considers the Mass as the greatest possible prayer of intercession insofar as it is the perfect offering of Christ to the Father by making present the paschal mystery of his death and resurrection."
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negative:
Suffering, tedious chores, etc.
Not knowing that much about it, I think of the concept of consecrating oneself, one’s family, to . . .
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Sacred Heart of Jesus (LINK)
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Immaculate Heart of Mary (LINK)
. . . where a person offers up themself to Christ, offers every breath they take, every beat of their heart to God (to God, through Mary). To me, offering up even ordinary things for a special intention seems to come from the same mind as consecrating oneself.
These are just examples; there are many other forms of consecration but these are the two I remember right away.