Thucydides notes that Homer recognizes the origin of the Hellenic identity to Achilles and his men, wherefrom all the other Greek-speaking races were named Hellenes. The etymology of the name 'Hellen' is not very certain, but it seems that it has a priestly and ascetic meaning. It is beyond doubt that the hellenic race is connected with Selloi, the priests of Dodona at Hepeirus (cf. Homer's, Iliad, Π 233).
The name Graecus (Γραικός - whence Greece) just as the name Hellen (Ἕλλην - whence Hellas - Ἑλλάς) comes from the mythical times of Deucalion (both are sons of Deucalion), and is connected with age (geraios -old- γηραιός), which is an epithet honoring the priests of Dodona. The name Hellen comes also from an epithet belonging to the priests of Dodona (Σελλοί - Selloi).
A part of that race went to Thessaly under the leadership of Achilles and with the name Hellenes (Ἕλληνες), while other parts went to the South and founded Graea (Γραῖα) in Boiotia and Euboia, or remained in Epeirus wherefrom, together with people from other Hellenic races, some of them went to Italy. Latin people knew Greece from them or from those who remained in Epeirus, and this is why they named all Hellenes Greeks (Graeci).
you made an imaginative etymology, which at least sticks to the Greek language.
(I'll use capital E for eta and O for omega)
Please compare gEraios and graikos: Firstly, it is most unlikely that a phonetic contraction would occur when the vowel between g and r is long; it is most likely to occur when the intermediary vowel is y; secondly, the rigid or consonantal letters of graikos are GRK, not GR. If you compare a noun and an adjective, you might say that -ikos is simply the adjectival ending, but here gEraios is already an adjective, with its own ending, -ios or something to that effect.
The adjectival substantive graikos refers to a person [any of certain inhabitants] who was likely named after his habitat or the land of his haunts. I don't find the base-noun in my classical-Greek lexicon, but we know the Latin version: Graecia; hence, * Graikia or something to that effect.
I looked for words with the etym "graik." No results. Probably the land was named after him, so that when I remove the adjective ending -ikos, I am left with the etym GRA. -- grastis = crabgrass; grass -- grastizO = I feed with grass -- graO = I graze; I take to pasture. [Latin: pascolo, in both senses of the term.] so, graikos = grazer; a human graikos = a pasture-man, a shepherd graikia = a pasture; grassland (grazer-land; shepherd's land).
Possibly the Hellenes of the grasslands or hinterlands, rather than of a country or state, were called Graikoi, in contradistinction to the Hellenes as sea-people or of the coastal lands. [The historical use of the term needs a separate investigation.]
I don’t understand why you call ‘my’ etymology of the name of Greece as ‘imaginative’. I don’t know what dictionaries you use, but Liddell-Scott explains graekus as coming from grais (old woman). Note also that your approach of Hellespont is surely imaginative, since Hellespont means the sea of Helle (Helle is the name of the daughter of Athamas, drown in that sea).
Sel- and hel- are connected as in selas and helios (sun), they have to do with illumination. Besides this (and regardless of any etymology) the whole historical information that we have, confirms the connection of the race called Hellenes with Selloi, priests of Dodona in Epirus.
I said " imaginative" because many ancient people used to attribute a race of people or the foundation of a country to some hero, priest, or god. As a rule, ancient etymologists relied either on some legends or on the resemblance of certain words. You see, a word like gEraios (with an eta or long e) is unlike to disappear by speakers (or learners) making a sound contraction. Usually contractions occur when there is a short vowel, especially a ypsilon. Anyway, does it not sound a little strange to say that a race of people [an "ethnicon" as I would say] would be named after "elders"?
Since you mention two places by the name of Graia (on which "graikos" is reasonably based), I would look for  some real evidence as to why they received that name, and  for some historical reason why "Graeci" rather than "Hellenes" became the general name in Southern Italy for the the Greek speaking colonizers ; that is, why the Roman used this name. The Greeks themselves in Southern Italy identified themselves from their place of origin, except for a group in Calabria where they called themselves Italiotes, after the local name of the region: Viteliu, which they wrote as Oyitalia [OY being the closest sound to V] or simply Italia. For strange historical reasons, including the Roman conquest of Southern Italy, they were generally called Graeci, while the name Italia was extended by the Romans from Calabria to near Rimini. (The little Rubicon River was the northern boundary of Italy [Italia] at the time of Caesar.)
There is a saying in southern Italy for people who have heated arguments or make a lot of verbal noise, "They look like the Greeks." So, I have been wandering whether this saying merely refers to a distinctive trait of the Greeks or, to begin with, was the reason why they were called "Greeks." I have to look into this.
You may have noticed that I removed my post on "Hellenes" and "Hellespont" since I realized I had made a false assumption. (I am very exacting on myself). It may be true that the Hellespont was named "the sea of Helle". It also true that Canaan may have been named after a grandson of Noah. However, genealogies do not explain the meaning of proper names... and all propers names had a meaning, at least once upon a time. So, we still do know what "hellene" named. And is the main etym of the word really "hell-"??? I myself hope to come back to this issue. ======================================================= Some findings:
My classical Greek dictionary lists geraios (with an epsilon), = old, venerable; elderly. This word was obviously contracted in ancient times, as the classical dictionary includes also: graios [= old]. Then there is: E grays [noun, feminine] = old woman; wrinkly thing [skin; film on boiled milk]. Graologia = old woman chatter. Graia [ADJECTIVE, feminine] = old.
Conceivably a city, region, or other feminine place could be called Graia, thus using the adjective as a noun ["The Old One"]. So, an inhabitant could be called Gra-ikos [Latinized as Graecus]. It is possible for this to have happened, in which case "Graeci" is a generalization of the people from the places you mention, but I would look for a historical corroboration of this hypothesis. At the same time, I would not associate "Graikoi" with the elders of Dodona, since here "Graikoi" would mean something like "the people of the elders." --- That Southern Italian saying must be an old one but not older than Roman times: probably the inhabitants heard "graikoi" when the Latin speakers on location used the word "gra'culi", which referred to crows or their sounds [gr, gr, gr]. (The Italian "gracchi," which has the sound of "grakki", refers to crowing sounds, to repeated crowings, and to crows. A woman is called a gracchia when she does a lot of chattering: "old crow" in English. ) So, it seems that, to the Southerners, "They behave like crows" sounded like "They behave like Greeks" [.... sicut graculi :: ... sicut graikoi].
The problem with HOMONYNMS [words or etyms which look or sound alike but have different meanings]:
When they exist in a language, and an attempt is made to trace back the etym of a word to any of the two more existing homonyms, it is impossible to tell which is the ancestral one. That is why some historical or other corroboration is necessary.
Thus far I have traced "graikos" back to two possible etyms: GRA:
(I) graikos graikos = gra + ikos gra = *gra in graO and other words (referring to grazing and grass).
(III) Then, having witten "sicut graculi :: sicut graikoi," I got the hint for another inspection: korax [koraks] = corvus; crow korax > kraO, = I crow; scream. (ATTESTED CONTRACTION!) So, possibly: graikos = gra + ikos gra < *gora The etym GOR is found in: -- gor-gos: fast, impetuous; (in Attic usage:) fierce, frightening. -- gor-gyra: subterranean aqueduct; Latin: gurges * ; English: gorge. -- Proper names: Gorgias (sophist from Sicily); Gordion (capital of Phrygia); GorgOn (Gorgon). -- gor-dyEnoi: inhabitants of Gordyaia in Armenia. ---------- Consider: GOR as the distinct etym, evident in GOR-gyra, since -- gyros` (adjective): round, curved. -- gyros (noun): turn; round hole [to plant a tree in]. -- Gyrai petrai , reefs [obviously round] at southern Euboia. -- gyrai < *gyra. So: GOR-GYRA minus the GYRA [referring to the canal-roundness of the gorge] leaves GOR = impetuous [as in a river] or frightening [in its depth or impetuousness]. GOR > *gor(a); Gor(gos); etc. *gora > *gra. Hence, Graia [city] = * The Impetuous. [Naming like Genoa: La Superba; Venice: La Serenissima; etc.] Graia [Goraia] > graikos [goraikos]; as a noun: "an Impetuous one." ================================================================= * The Italian derivative from Latin gurges/gurgitis, namely gurgito [= gorge], has become obsolete; "gola" is used instead. However, the verb "rigurgitare" [= to regurgitate] is in effect. But the meaning of regurgitating [food] is secondary; the primary meaning is that of "gurgitating backwards," that is, filling and swelling and then VIOLENTLY thrusting backwards. The notion of forceful or impetuous drive --GOR -- is central, without any connotation of location or shape of location.