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Modern Spanish's hay has the following etymology:
From ha, third-person singular present of haber (“to have”), + i, from Latin ibī (“there”).
However, the other tenses' equivalent of hay, such as 'había' and 'habrá', have no such trailing 'y' or 'i'.
Why is this, do they have separate etymologies?
I agree with CarlosAlejo with regard to something being there in the present tense. I found further reference about the origin of the verb and the modifications it suffered in the interesting volume Morfología del verbo castellano by Rufino Lanchetas, which was published in 1897 but is nevertheless full of useful information. Regarding the verb haber it says, among other things, this:
Indicative present.— First person.— In relation with its Latin origin habeo and the uniformity of the radical, and according to the seventh phonetic rule, it should be habo. This form is not found in any period in the history of Spanish , and we should look elsewhere to expain it. The first person, habeo, as well as Subjunctive habeam and habeas had, from the time of the Gracchi, the e in the formative suffix changed into an i, thus instead of habeas and habeat we find habias and habiat [...] So [...] the Latin form habeo first became habio. With the b lost as per the sixth phonetic rule, we should have got haio, as in habias = hayas. This form is likewise not to be found in Neo-Latin languages — we can only say that for reasons unknown the o was dropped, thus giving us hai, a form that is found in Provençal, French and Asturian. Hai became hei, as retained in Portuguese and Galician. Spanish went one step beyond and, as in sei = sé and amei = amé, went through hei = * hee and came from a very early time to establish he both in the present tense as well as in the future derived from it. Phonetic process: habeo, habio, * haio, hai, hei, hee, he.
Subjunctive.— Every form in this mood can be explained by the same process used for the first person of the Indicative present, with the peculiarity that the forms in Subjunctive present were limited to the loss of the b and the conversion of the palatal vowel i into a consonant. Their phonetic process has been as follows: habeat, habiat, habia, haia, haya, and the same in the other persons. The form haya is very old: "Desto que ellos fablaron nos parte non ayamos." (Poem of El Cid.) [...]
The Indicative third person singular present gets a paragogic y when used as unipersonal [impersonal?] meaning "to exist" or "to have", e.g. "Hay Dios." Its origin may be traced back to the 15th century when some monosyllabic forms got the addition of the mentioned paragogic y, such as soy, voy, doy. [Could this conflict with the etymology of hay as haber + ibi?]
Second group: past imperfect (había, habías, habíamos, etc*) / future When forming these tenses based on the Infinitive, it would be expected for them to be haberé, habería. We can sometimes find these full forms in ancient documents [...] [b]ut the most usual form at all times was the current one, with a syncope: habré, habría. [...] This syncope can be explained in the same way as that in sabré and cabré. It consists of the supression of the forming suffix e by the accent in the final e, something helped by the former being unstressed and the resulting consonant group bre being easy to articulate. Portuguese and Galician retain the full form haverei, haverás, etc.
Certainly a lot of time has passed since this was published and perhaps it would be wise not to take it as the absolute truth, but I found no newer sources as yet.
After some more googling, I found some more material:
The 1948 article Una pesquisa acerca del verbo haber, hosted at the Centro Virtual Cervantes, says that
The phonetic evolution of this verb has been perfectly researched and its evolutionary movement from the Latin form to the present Spanish one can be traced step by step. There can be neither discussion nor doubts about it, and it is enough to read Menéndez Pidal's Gramática histórica to satisfy the curiosity of even the most exigent reader.
The article, a discussion of the impersonal use of haber, goes on to say:
In the Poem of El Cid the verb is scarcely found accompanied of the adverb y or i or hi, and sometimes without it, used as an impersonal or unipersonal verb. In such a long poem it only appears a couple of times:
Entre Minaya — e los buenos que i ha [...]
Todos eran ricos — quantos que allí ha [...]And the classic example found in every explanation of the certainly scant history of the use of haber as an impersonal verb:
Tales i a que prenden — tales i a que non.
I looked for Menéndez Pidal work and found another very old edition of the Manual elemental de gramática castellana, which says in the section concerning the Irregular present:
In the past, habere presented some forms derived from Classical Latin habes (but not from the 1st person habeo) and habet: "ave", "aves", "avemos", "avedes", "aven". But the ones that prevailed were derived from a contraction this verb suffered in Vulgar Latin in order to make its use as an auxiliary easier, in which the stressed vowel and the suffix: 1st person haio, "heo" or "hey" (antiquated) and "he"; 2nd person, has, "has"; 3rd person, hat, "ha", which when joined with the adverb "y" gave the impersonal hay; 4th person, (hab)emus, "hemos"; 5th person, (hab)etis, "hedes", "heis"; 6th person, hant, "han". These are the forms that prevailed, duplicating the 4th person with "habemos" and reserving the 5th person for use as an auxiliar ("amar-eis" and Classical "heis de estar", etc.), using "habeis" for the other cases.
Subjunctive. Classical habeam is clearly reflected in Asturian dialectal eba, ebas, etc. The Vulgar Latin contraction hajam, -s, -t is the one that gave the usual form "haya", "hayas", etc.
Imperative. habe, "ave", which was still used in Spanish Classics; "habed", not used very often nowadays.
Sadly, I do not have access to modern material such as Ralph Penny's Gramática histórica del español, and actually know very little about Spanish historical grammar, but it looks then like it is rather settled that the impersonal hay is indeed the fusion of ha plus an adverb derived from Latin ibi or hic. With regard to that, further searching led me to an article in the much more recent book Cuestiones de actualidad en la lengua española that says that
The addition of -y in the 1st person Present in the irregular verbs dar, estar, ser and ir is a topic of eternal discussion in Spanish historic morphology. There are at least four hypotheses about its origin: analogy with the ancient form "hay", "hey" from habeo; agglutination of a pronominal subject ("so yo" > "soy"); agglutination of the old adverb "y" from ibi/hic as in "ha" + "y" > "hay"; and addition of a paragogic vowel ("soe" > "soy").
From what I have been superficially looking at, not many scholars have tackled the question — one who did was Joel Rini in his book Exploring the Role of Morphology in the Evolution of Spanish, which has a section on "the -y of Spanish hay". Basically, he questions current theories and points out that the -y appears only in present tense and is a suffix instead of being in front of the verb, unlike in French, Catalan and Italian (il y avait, hi havia, ci era), and also gives examples in which the joined form ay is used together with the adverb y, such as
Et daquel Jnfant Theseo que fue muy buen uaron & de muchos et muy grandes fechos que fizo Egeo esse Rey de Athenas uos contaremos adelant en sos logares de muchas & grandes razones que ay y dellos. (Alfonso X, General Estoria)
He goes on to argue that the origin of hay might be found in the subjunctive haya and not in the addition of the adverb y to ha. On the other hand, he points out that researcher Érica C. García argued that
[...] the agglutination of y occurred only in the present tense because of the much higher frequency of the existential use of haver in in this tense: "The reinterpretation of aver + y took place, however, only in the Present Indicative — far and away the most frequent tense for the existential use of the verb, especially in spoken language."
A long article (in Spanish) about the state of the question of the final y ("Doy, estoy, hay, soy y voy: cinco monosílabos con una terminación extraparadigmática. Estado de la cuestión.") can be found in the book Estudios de historiografía lingüística.
I have two possible theories I can't prove, just from my own imagination.
Starting from the etymology of hay: ha + hi → "ha y" (literally he has there). The first theory is that other tenses also used the trailing hi, but only the present tense ended up incorporating the trail into the form and generating ha i and then hay because of the similarities in the pronunciation of the words. The other tenses just suppressed the trailing hi or simply didn't incorporate it because there was no cacophony. This could be from a similar (but not the same) process that generated soy, doy, estoy and others.
The second theory is that if you think about it, you only can know that something is in a given place ("there") in the present tense. You are seeing an object there and thus you can say there is (ha hi). In other tenses, e.g. había, hubo, habrá, you cannot be sure that the object was (or will be) there, so you couldn't say había hi or habrá hi.
I'm sorry I can't prove these theories, but I hope this will give an starting point to investigate further. Nonetheless, if I discover something else I'll come back and add it here.
Another clue: this is written in the Cantar de mio Cid (circa 1200):
Catando estan a Myo Çid quantos ha en la cort,
a la barba que auie luenga e presa con el cordon.
So back then you could say "ha" in the sense of the nowadays "hay", so the creation of the latter must have happened after that.
