The only free and comprehensive online etymological dictionary of the Spanish language
haber (Verb) "to have"

12th cent. Old Spanish aver. From Latin habere "to have," "to hold." From Proto-Italic *χab-ē 'id.' From Proto-Indo-European *ghh1b-(e)i̯- 'id.' The phonology of this word is exceptional as the presence of *b was rare in Proto-Indo-European.

Romance: Western Vulgar Latin: Asturian haber, Portuguese haver, Galician haber, Catalan haver, French avoir, Italian avere; Eastern Vulgar Latin: Aromanian amu, Romanian avea; Sardinian: àere

The spelling change from aver to haber was a modern modification to more closely reflect the original Latin.
habiente (Adjective) (law) "possessing"

Borrowed from Latin habientem, accusative of habiens 'id.' From habere "to have" (see haber).
habitación f. (Noun) "habitation"

13th cent. From Latin habitatio, past participle of habitare "to inhabit." From habere "to have" (see haber) and frequentive suffix -itare (see note under faltar).
habla f. (Noun) "speech"

12th cent. Old Spanish fabla. From Latin fabula 'id.,' from fari "to talk" and instrumental sufix -bula (see -ula). Latin fari is from Proto-Italic *fā- "to talk." From Proto-Indo-European *bheh2- 'id.'
hablar (Verb) "to talk (with/to)"

12th cent. Old Spanish fablar. From Latin fabulari "to chat," from fabula "conversation," "talk" (see habla, fábula).

Romance: Western Vulgar Latin: Asturian falar, Portuguese falar, Galician falar, Italian favellare; Sardinian: faeddare
hacer (Verb) "to make," "to do"

11th cent. Old Spanish fere. From Latin facere 'id.' From Proto-Italic *fak- 'id.' From Proto-Indo-European *dhh1-k- 'id.' From the root *dheh1- "to put" (see dar).

Dialect Variants: Navarre fer, La Rioja fer, Aragon fer

Following Meyer-Lübke (1890), Old Spanish fere and its dialect variants in fer point to a stress variant in Vulgar Latin *facére that reduced to *fére. Modern Spanish hacer reflects Vulgar Latin *fácere > *fácer and then remodeled to hacér in the Middle Ages. While an argument from stress variation in Vulgar Latin should raise eyebrows, it has at least one strong case in pero / peró.

A further case that Vulgar Latin *fácere existed is in Old Catalan far "to make," from a pre-form *fare that must derive from a stress-heavy /a/ that caused syncopation of the interior -e-. See Rufat (2013) for further discussion from a Catalan perspective; Chambon (2013) for additional remarks.
hacia (Preposition) "toward"

12th cent. Old Spanish faza; earlier faze a. From Latin facies ad "facing toward." For the etymology of facies, see haz (1); for the etymology of ad, see a.
hambre f. (Noun) "hunger"

10th cent. Old Spanish fambre, earlier famne. The n in famne was tapped by the tongue and became *famre. *b intervenes between *mr via epenthesis to become fambre. From Vulgar Latin *famine 'id.,' from Latin fames 'id.' Of unknown origin. A connection to *dhH- "to disappear" has been proposed, albeit with its own problems. See Walde & Hoffman (1954) for further discussion.

Romance: Western Vulgar Latin: Asturian fame, Portuguese fome, Galician fame, Catalan fam, French faim, Italian fame; Eastern Vulgar Latin: Aromanian foami, Romanian foame; Sardinian: fàmene

Words for hunger tended to be borrowed into Indo-European languages or internally innovated. "There is only one word reconstructed in Proto-Indo-European that means 'hunger' (a Hittite-Tocharian isogloss) and even this is problematic in tha a comparison between Hit kāst- 'hunger' and Toch B kest 'hunger' still only yields a PIE *Kos-t-, i.e. we can only say that the word begins with a velar but must be uncertain which velar that is (it could be *ges- for example)." ~ Mallory & Adams, The Oxford Introduction to Proto-Indo-European and the Proto-Indo-European World (2006)
hasta (Preposition) "until"

13th cent. Old Spanish fasta. From Arabic ḥattā 'id.'

Borrowing of prepositions is a rare phenomenon. Corriente (1983) argues the act of borrowing was influenced by the Latin phrase ad ista "to this." For the continued etymologies of these words, see a and éste respectively.
haya (1) f. (Noun) "beech," "beech tree"

14th cent. From the second element of Latin materia fagea "beech wood." From fagus "beech tree." From Proto-Italic *fāgo- 'id.' From Proto-Indo-European *bheh2g-o- 'id.'

Also the origin of the surnames de la Haya, Hay, Hayas, Faya, and Fayos.

Romance: Western Vulgar Latin: Asturian faya, Portuguese faia

Indo-European: Celtic: Gaulish *bāgos "beech;" Germanic: Old Norse bók "beech," Old High German buohha 'id.,' Old Saxon bōka 'id.,' Old English bōc 'id.' (English beech); Hellenic: Ancient Greek φηγός (phegós) "oak," "acorn"