Alaikumassalam wa rahmatullah,
Asaf, pronounced [(A)fter] + [(SU)n + (F)lower], is said to have been a adviser or scribe of Prophet Sulaiman, whose full name was Asaf bin Barkhiyaa. Many sources wrongly pronounce the name as Asif, but the correct pronunciation is Asaf, with emphasis on the first syllable, as mentioned by the scholar al-Firoozabadi. Asaf bin Barkhiyaa is said to have been the person responsible for carrying the throne of the Queen of Sheba to Prophet Sulaiman in the blink of an eye.
The name of this person or his details are not mentioned in the Quran or any of the sayings of the Prophet. The only reference to him is in verse 40 of chapter 27 of the Quran:
The one who had knowledge from the Book said, "I can bring it to you before you even blink your eye." When Solomon saw the throne placed before him, he said, "This is a favor from my Lord by which He wants to test whether I am grateful or ungrateful. Whoever thanks God does so for his own good. Whoever is ungrateful to God should know that my Lord is Self-Sufficient and Benevolent."
The mention of the name Asaf is in some words of Ibn Abbas quoted by an-Nasaa'i, and it is uncertain whether this is authentic or not. The name itself is of unknown origin, it might not be Arabic, and since the name is not mentioned in any of the authentic sources, we cannot say it is an Islamic name.
You can use the name Asaf since it doesn't have a known bad meaning, but we don't know if it is truly the name of the adviser of Prophet Sulaiman.