OED confirms that it is from Tibetan sharpa, inhabitant of an eastern country. The other forms listed are Serpa, Sharpa and the earliest forms recorded are Serpa and Sérpá from 1847.
1847 B. H. Hodgson in Jrnl. Asiatic Soc. Bengal XVI. 1237 Cis-Himálayan Bhotias vel Tibetans, called..Serpa, &c.
1847 B. H. Hodgson in Jrnl. Asiatic Soc. Bengal XVI. 1238 The sub-Himálayan races..inhabit all the central and temperate parts of these mountains, the juxta nivean or nethermost tracts being left to the Rongbo vel Sérpá.
The following excerpt is from an article titled "Sherpa History" from the official site of Tapting (a village development committee in Nepal) and explains the origin of sherpa with historical references and citations from ethnologists.
Sherpa is an easterner indigenous people, live in the eastern part of Nepal. The word Sherpa is combination of two words – "Shyar" (East) and "Pa" (people) "People of the East" and the words are coined from the Tibetan language.
The word 'Sherpa' means easterner as it is addressed to the present day Sherpa. It is not clear how this term came to be associated with this particular group
Previously they were not addressed by the word 'Sherpa'. But it is not clear in what circumstances this term came to be associated with this particular group from the Tibetan point of view, Sherpas dwelling in the highlands of Nepal are southerners rather than are easterners, though there is the tradition that before their migration to their present habitat they had settled in eastern Tibet (Haimendorf 1964,1984)
In this context, some Sherpas came to the conclusion that they address 'nubripa', western, for the same type of culture group whose habitat is western part of Nepal. Similarly, the term Sherpa is used for the "eastern inhabitants of Nepal".
Another group opined that the term indicates" man from rising sun area". Therefore, they were addressed Sherpa. But it is so why the other ethnic groups of Eastern Nepal is not addressed by the term "Sherpa".
According to reincarnated Lama of Tengboche monastry, the first man came to Khumbu from Kham Salmo-Gang (east of Tibet) whose clan was called Timi (Thimi), incarnated from the god Wosal and his name was pajzin. After he returned back to Tibet the people asked him where he came from he replied "the east part of Kham" That is how the name Sherpa, easterner came into being (Kunuwar,1989)
But some Sherpas of Solu are not in favor of this opinion. According to them, the word Sherpa is of very late origin.
It has come to know on the basis of different literatures cited above that Sherpas were migrated to Nepal from the Kham region of Tibet 5 to6 hundred years back.
It is presumed that the group of people migrated out from Kham region, east of Tibet, was called as " Shyar Khamba" (People who came from eastern Kham) and the place where the migrant people started to settle was called " Syar Khumbu". As the time passed the " Shyar Khamba", inhabitants of shyar Khumbu, were called as Sherpa.
However, I did some research regarding your theory about sher being lion in sherpa and I could find some hints related to Thakali language (Thakali is one of the ethnolinguistic groups in Nepal along with Sherpa and these two groups share the same region in Nepal). (see map).
The first reference I could find is about the Thakali clan Sercan which has a lion on their emblem; and the root of the word is mentioned as ser- which suggests that it might mean lion. (from the book Himalayan Anthropology: The Indo-Tibetan Interface
edited by James F. Fisher):
However, one cannot fail to compare the Thakali Sercan (emblem: the white lion of the glaciers) with the Sherpa Serwa, especially as -can is a suffix. The accepted etymology of Sherpa is Tibetan shar [east] (Snellgrove 1957:215; Oppitz 1968:49), but one suspects at the very least contamination with the root Ser-; spoken Nepali has only one sibilant phoneme.
Note: Snellgrove is a tibetologist and Oppitz is an ethnologist.
I continued my research by focusing on the term Sercan and I could find another form of the term which is Shercan. It is one of the four Thakali clans and the clan protector is mentioned as the white lion of the glaciers. (from the publication Ritual practice and group maintenance in the Thakali of central Nepal by Andrew E. Manzardo)
Then, I could finally find a source that actually says sher means lion in Thakali language. Here is the relevant excerpt from the book The Thakali: a Himalayan ethnography by Michael Vinding:
Since the Nepali clan names were introduced in the present century, it is somewhat surprising that the Tamang Thakali do not know their meaning. The account of the Cyogi clan, the Cyogi Rhab, written in the Devanagari script, renders the clan names as gaucan, tulacan, syercan and battacan. The ending -can may be an attempt by the names inventor(s) to connect the Tamang Thakali with the Thakuri clan Chand. Gau means 'cow' and may be related with gaucan clan god. Sher means 'lion' and appears related with the Shercan clan god. Bhatt_ may mean 'exchange' and may be indirectly connected with the Bhattacan clan god. The meaning of tula is a puzzle.
On the other hand, lion is senggi in Sherpa language (from Sherpa-English English-Sherpa Dictionary by Nicolas Tournadre, Lhakpa Norbu Sherpa, Gyurme Chodrak & Guillaume Oisel):
Oh! Let's not forget pa. This Tibetan affix strongly suggests that it is related to people in the word sherpa, although it is a very versatile affix that can be used in a variety of ways. The below excerpt is the definition of pa from A Tibetan-English Dictionary with special reference to the prevailing dialects By H. A. Jäschke:
པ pa, an affix, or so-called article, the
same as ba (q.v.) which, when attached to the roots of verbs, gives them the signification of nouns, or, in other words is the sign of the infinitive and the participle; in the language of common life, however, it is frq. used for the finite tense, and for par; affixed to the names of things, it denotes the person that deals with the thing (rtá-pa horseman, čú-pa water-carrier); combined with names of places, it designates the inhabitant (bód-pa inhabitant of Tibet); with numerals, it either forms the ordinal number (ynyis-pa the second), or it implies a counting, measuring, containing (bú-mo lo-ynyis-pa a girl counting two years, i. e. a girl of two years; Kru-gánpa measuring one cubit; súm-cu-pa containing thirty viz. letters, like the Tibetan alphabet); frq. it has no particular signification (rkéd-pa etc. etc.), or it serves to distinguish different meanings (rkan marrow, rkán-pa foot) or dialects (ká-ba B., ka W. snow); pa dan with a verb, v. dan 4; in certain expressions it stands, it would seem, incorr. inst. of pai: ysó-ba rig-pa science of medicine, grub-pa lus structure of the body, dám-pa čos holy doctrine (of Buddha).
In conclusion, the accepted etymology of sherpa is Tibetan sher (east) and pa (people or inhabitant) but there is a good evidence that sher might be related to lion based on the above findings.