Assalaam Alaikum wa rahmatullah,
It is a blessing to make supplications for other brothers and sisters. Consider this Hadeeth of the prophet (SAWS): The Prophet (saws) said:
Whenever you make a supplication for another believer and he is not present, an angel will say ‘and same to you.’
Allah Subhana wa ta’ala has provided us the opportunity to dramatically increase the effectiveness of our own Dua’s by encouraging us to make dua’s for other brothers and sisters in Islam. In a Hadith the Prophet (saws) said:
“The supplication that gets the quickest answer is the one made by one Muslim for another in his absence.”
And your Lord said: “Invoke Me, (i.e. believe in My Oneness (and ask Me for anything) I will respond to your (invocation)…………..” (Quran: 40: 60)
It is mustahabb for us to ask Allaah for what we need by name, using whatever simple words and phrases Allaah enables us to use.
Thus, if the brother / sister did not specify what the dua is to be about, all you can do is say something like the following:
'Oh, Allah, please grant (person’s name) his/ her dua if it is good for him/her and withhold his/her request if it is bad for him/her. Your knowledge is the best of knowledge and thus, always grant us the good for which we ask when there is good in it in this world and the hereafter.'
This is just a suggestion, as duas are personal according to personal needs. The wording should reflect that what is being sought from Allah Subhna wa ta’ala.
It is also very important that we do not just rely on others to make a dua for us. We must ALWAYS ask Allah for ourselves and ask Him directly when praying and at other times prescribed in the Sunnah.