I am by no means an expert and so this answer may be frowned upon, but based on my, admittedly light, reading on the subject over the last hour or so, Abyssinia is generally accepted as an evolution of the Arabic (though from what I've read, it may also have developed from one of the South Arabian languages) Habashah (حبشة). My dictionary (Hans Vehr, 4th Ed.) only has definitions for that as Ethiopia/Abyssinia or the Ethiopian people or language but most of the things I've read state that Habash is used to describe the people now while Ethiopia (Ityioppia) is the name of the state.

I am having trouble finding Arabic-language sources that discuss the etymology of the word Abyssinia but I did find a reference to Stephanus of Byzantium referring to the people as 'Abasēnoi' (Αβασηγοί) which would mark the evolution from Habashah to Abyssinia (the H is dropped and Greek ending -noi added yielding Abyssinia as it spreads to other languages).

According to The Etymological Compendium by William Pulleyn (p. 270), the meaning of Habashah is something akin to "mixture" which I have read in other places as well. This falls in line with it referring to the peoples rather than a state. Pulleyn also calls Ethiopia an "epithet" given by Greeks to "all countries inhabited by Blacks." And if you dig into the etymology for Ethiopia you will generally find it to refer to skin of dark complexion where 'aitho' is 'to burn' and 'ops' is 'face'.

I have seen pg. 948 of the Encyclopaedia Aethiopica: D-Ha referenced several times in relation to the etymology and history of Habash/Habashah but unfortunately I do not own this book and the preview for it on Google Books does not include that page or the page that follows. If you or someone else owns this book, that could help shed some additional light.

I would really love for someone with a greater background of study in this area or access to more/better resources than I to fill in some of the blanks but as near as I can tell, Abyssinia has older roots than Ethiopia however, the people are sometimes still called 'Abyssinian' in Ge'ez and Arabic while referring to the state as 'Ethiopia' (though I can attest that some Arabic sources still use the older Habash when referring even to the state).

The Arabs called it Habasha. A group of Muslims went there to escape persecution in Mecca, seeking (and obtaining) protection of the Christian King.

Im Ethiopian and Im from one of the "Abyssinian" ethnic groups. Theres been an ongoing debate amongst Northern Ethiopian intellectuals about the origins of the term Abyssinia but we know it is a derivative of the term Habesha which encompasses most semitic Northern Ethiopian and Eritrean ethnic groups. Some think it was derived from the name of a Yemeni tribe called the Habeshat who settled in the Ethiopian/Eritreans highlands during the Dmt era. Many think its an ethnic slur given by the Southern semites to Northern Ethiopians and Eritrean (who were heavily engaged in Arabian politics at the time under the Aksumite Empire which expanded into what is now Yemen and Oman) which basically meant mongrel race or darker Arab in an extinct south semitic language. All in all, most agree we will never know for sure.

The term Habesha and its English equivalent Abyssinia has significant political and social implications in Ethiopia and Eritrea right now so I suggest you not use that term publically, especially amongst older Eritreans. Its a controversial term to some but its also a unifying term for others that symbolises the historical achievements of the Abyssinian people of Eritrea and Ethiopia and our shared culture, history and origins. But I notice many younger Ethiopians and Eritraians in the west unaware of the history of the term use it as a racial identity which i find very strange.

An example of our shared traditions and culture is shown in our clothes and hairstyles. These pictures are from three of the Habesha ethnic groups that occupies the Highlands of Ethiopia and Eritrea. They all look identical http://m.imgur.com/gallery/efXr4