How do Jehovahs Witnesses understand Jesus inheritance of Gods name Hebrews 1:3-4 reads: 


  the Son is the radiance of God’s glory and the exact representation of
  His nature, upholding all things by His powerful word. After He had
  provided purification for sins, He sat down at the right hand of the
  Majesty on high. So He became as far superior to the angels as the
  name He has inherited is excellent beyond theirs.


The last phrase is particularly interesting, it says that the Son (Jesus) inherited a name. Notably this can't simply mean that Jesus was given a name, but that Jesus was given a name that belonged to someone else. This clearly can't be a name of one of the angels - because it specifies that "the name He has inherited is excellent beyond theirs" - and it makes little sense for it to be a human name.

My understanding is that Jehovah's witnesses consider Jehovah, to be the greatest name, and the name of God. In this context the only explanation I can see for Jesus being a given a name more excellent than that belonging to the angels is that he is given this name the name of God. This makes sense because if he is inheriting it, he also has to be receiving the name that belonged to the one that gave it. 

This makes sense of several passages. For example in Phillipians 2:9 it is written:


  Therefore God exalted him to the highest place and gave him the name
  that is above every name,


If from a Jehovah's wittiness perspective, Jehovah is the greatest name, these passages make sense together. Jesus is given the name Jehovah.

Further this explains why the new testament writers put such a significance on the phrase "Jesus is Lord"  (1 Corinthians 12:3; Romans 10:9; Acts 8:16; 19:5 and 1 Cor 6:11) (notably when the bible was translated into greek the divine name, translated by JWs as Jehovah, is rendered kyrios, the word translated as Lord in these passages). When the apostles and bible writers use this phrase they are affirming that this inheritance of the divine name has taken place.   

If this isn't the conclusion that JWs reach, how do they understand these passages?

EDIT:
In the early church there was clearly debate about the relationship between the Christ and his Father - clearly within the lifetimes of the apostles some thought that Christ wasn't divine, and people close to the apostles did (Polycarp, Justin the Martyr, and Ignatious). From a JW perspective, why do people think the apostles used this kind of ambiguous language? 


Talking about Jesus "inheriting"/being "given" what appears to be the divine name.
Identifying Jesus frequently as kyrios (i.e. the Greek word that YHWH was replaced with when the Septuagint was translated into Greek, commonly translated as Lord in english bibles).


Was it for example, to avoid refuting the early Christians who believed in the divinity of Christ and thus maintain unity in the church? 



[the mainstream christian reading of the bible is that YHWH (and its Greek and English circumlocutions kyrios and Lord) is a name/title shared by the Father, Son and Spirit. It's beyond the scope of this question to discuss what this means in the context of Heb 1 but in the is post I am proposing that the most natural reading of the text is that the writer of the book is discussing that sharing via the metaphor of inheritance, much as the author of Philippians discusses that sharing by metaphor of a gift.]
There's a 1984 Watchtower article which comments on this pair of scriptures. The conclusion drawn is that the &quot;name&quot; which Jesus inherited is his position, or reputation, as King. This is the definition of &quot;name&quot; that people refer to when they talk about &quot;making a name for yourself.&quot;

True, the apostle Paul wrote to the Hebrews: “He [Jesus] has become
better than the angels, to the extent that he has inherited a name
more excellent than theirs.” (Hebrews 1:4; Philippians 2:9, 10)
However, this describes his situation after his having been here on
earth. He was still the archangel and “the beginning of the creation
by God.” (Revelation 3:14) But he became better than the angels. The
‘more excellent name’ or position is something he did not possess
before coming to earth. (These scriptures contradict the Trinitarian
concept that the Son is and always has been equal in every way to the
Father.)
Hence, the fact that Michael is the archangel, chief of the angels,
the fact that he stands up to rule as King, and the fact that he takes
the lead in casting Satan out of heaven at the time of the birth of
God’s Kingdom all lead us to just one conclusion: ‘Michael the great
prince’ is none other than Jesus Christ himself. —Daniel 12:1.
— w84 12/15 p. 29