There are various short prayers (however, I have them only in Serbian in my prayerbooks) for other people: family, friends, enemies, compatriots, people who asked us for prayers etc. They're included into the morning prayers, and/or to the prayeres before the Eucharist, or into the section "other prayers"/"prayers for various occassions".There
is a nice interecessory prayer, that I have also in my Serbian prayerbook.
If it's a certain person, you can say troparion or even the canon to her/his patron saint.
There are also special canons and akathists for various situations (illnesses, alcoholism, depression etc.) in different languages, even in the Internet, you have to try with Google.
The paraklesis canon to the Theotokos I think it's good not only for an individual, but in geenral for people needing a spiritual or physical help.
Another, very simple option, is just saying the Jesus prayer in modified way: Jesus Christ, Son of the God, have mercy on x/them/him