I think the answer largely comes from the context. As far as the etymology is concerned, the word κεκληρονομηκεν basically just means "having inherited."* I think this starts to make sense as we look at the context.
Essentially, the author of Hebrews is starting out the book by defending Christ's authority as the Son of God. Hence the usage of words referring to inheritance. One of the primary differences of sonship as compared with servants is that sons were entitled to an inheritance (see Gal. 4:1).
When the author of Hebrews wants to point out that Christ is the Son of God and not simply an angel or some other kind of servant, he decides to present proof, and using an inheritance as proof is probably the most straightforward. The natural question then is, "Why did he choose a name as the inheritance?"
The answer is (at least in my understanding) two-fold: first, because the whole passage revolves around proving that Jesus is the Son of God, and secondly, because just as in the English, the word "name" (ονομα) can be broader than simply "what someone is called." It can also refer to their reputation, their fame.
Since the world is transient, to give Christ the inheritance of the kingdoms of the world or something of that nature would be transient as well, and consequently a poor defense of His Sonship.** To ascribe to Him the inheritance of Heavenly treasures and dominion would be valid, but disconnected from the readers. So the author of Hebrews instead speaks of an inheritance which has present consequences and continues into the future, ie., the name of Son of God, and the reputation or fame of worthiness of worship.
Being made so much better than the angels, as he hath by inheritance obtained a more excellent name than they. For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son? And again, when he bringeth in the firstbegotten into the world, he saith, And let all the angels of God worship him. And of the angels he saith, Who maketh his angels spirits, and his ministers a flame of fire. But unto the Son he saith, Thy throne, O God, is for ever and ever: a sceptre of righteousness is the sceptre of thy kingdom.
(Hebrews 1:4-8, KJV; emphasis added)
*I should probably mention that my Greek training is largely classical, so there is perhaps a bit of wiggle room for Biblical Greek scholars to take me to task. That said, I checked a few NT lexicons like Thrayer's to verify my understanding.
**Depending how you interpret words coming from Satan's lips, Jesus' temptation in the wilderness kind of shows that being given the nations of the earth is not a proof of Sonship (Luke 4:6)