This movement from Mozart's Requiem is known as either "Lacrimosa" or "Lacrymosa" (see for instance the Wikipedia article, which uses both spellings).
Why is there two different spellings and which one is considered correct (if any)?
I believe lacryma is generally considered a hypercorrect misspelling.
The archaic Latin spelling was lacruma, still sometimes used in classical Latin, or an even older dacrima/dacruma. The standard spelling was lacrima.
In Greek, it is spelled dakru or dakruon, which would be 'properly' translitterated into Latin as dacry(on). But the only spellings with a d- found in archaic Latin seem to be dacrima/dacruma, while Greek had no form on -ma¹. This means that it was not originally translitterated from Greek, and the letter y is only used in Latin words translitterated from Greek. The word may or may not have been borrowed from Greek in prehistoric times (there is no consensus¹), but that probably did not involve a y.
I believe the letter y came to be pronounced like i in many places at some point (probably after the classical age), as in modern French and Spanish. At that time, a common spelling error was to spell words originally written with a y as i, based on pronunciation. If writers thought that lacrima/dacrima was originally from Greek, they might have been afraid of making this mistake and hypercorrectly spelled lacryma.
The spelling lachryma is also observed, which may also be based on the hypercorrect assumption of a Greek spelling lachru-, for -ch- in Latin is normally limited to translitteration from Greek chi. At least I can't think of any other reason, even though though Greek has no letter chi in its forms of dakru(on).
¹) Michiel de Vaan, Etymological Dictionary of Latin and the other Italic Languages (Leiden 2008), s.v.
I'm not sure about the particular history of Lacrymosa/Lacrimosa and its derivation from dáḱru- (as @brianpck points out), but Vox Latina (by W. Sidney Allen) explains that words with i/y alternative spellings can reflect later changes in Greek pronunciation where upsilon and iota get confused (pg. 53):
In the popular Greek speech of some areas from the second or third century A.D. onwards υ had become confused with ι; consequently some borrowings into late Latin are taken over with i rather than y, and the spelling of earlier loans with i becomes common. This development is censured by the grammarians (e.g. Probus, 'gyrus non girus'), but is normal for such words in Romance (Italian girare, French girer, etc.).
The Greek upsilon (υ) can be rendered either with an i or y: and, in fact, the y was referred to as an “i Graeca”, as it still is in French. The two are both valid spellings, though a quick frequency check seems to indicate that the i spelling is much more common.
The above L&S entry, for instance, only has one instance (quoted from Virgil) of the y spelling.
the word Lacrymosa, generally, enters in the common language during the middle age when the Italian language was already well developed, in fact this is mostly an hybrid between the Italian word Lacrima and the latin Lacruma. In the case cited by the person who opened the topic we are talking about the Catholic sequence "Dies Irae" composed by Tommaso da Celano that contains a verse that begins with the "Lacrymosa dies illa ..." in the 14th century when the Italian language was already in high phase of development from the Latin and in which the purity of the Ciceronian times was just a faint remembering.
Since we are talking about the end of the eighteenth century (that’s when Mozart supposedly composed it), I will argue that the evidence already mentioned coming from etymology or Classical Latin spelling practices is not quite relevant here.
David Butterfield succinctly summarizes the development of Latin after the late medieval period (i.e. Neo-Latin or New Latin) in two words – “purification” and “decline” (Butterfield 2011: 303).
The defining feature of Neo-Latin is that humanist scholars dismissed Medieval Latin as “Latina barbara” and the only “correct” way to acquire Latin was through imitation (i.e. by reading Classical Latin writers). Naturally, such a daunting task was impossible to accomplish and, ironically, the language of Neo-Latin writers does not precisely follow the rules and practices of Classical Latin writers – see Butterfield 2011 for further detail (esp. pp. 310-313).
Now, about Neo-Latin spelling practices. To slightly paraphrase Butterfield, despite “the closer attention paid better to manuscript sources for ancient literature” and – hence – “an improved understanding of orthography,” “correcting the spelling of Latin to Classical standards was not a central area for scholarly attention” (p. 312). That is why we can find different spellings of the same word among Neo-Latin writers – it could be even the same author using different spelling variants in the same text! That is why Milena Minkova argues that “Neo-Latin orthography is not a strictly definable concept” (Minkova 2014).
Now, about the word “lacrima” (and its derivatives). De Vaan argues that TLL does not provide any information re: the distribution of variants lacruma and lacrima in Classical Latin, and I haven’t looked into this further – yet.
Peter Stotz (Stotz 1996) mentions, in the magisterial fifth-volume “Handbuch zur lateinischen Sprache des Mittelalters” (we need the third volume – “Lautlehre”), that in Medieval Latin there was the following rule:
“omnia Graeca nomina, que in usum Latini semonis veniunt, per ‘y’ scribenda sunt.”
This rule was generalized even to cases where ‘y’ shouldn’t be (it’s not etymological), e.g. the ending –ia in phylosophya [sic!]. This is known as hypercorrect spelling.
So, in this case the "correct" spelling would be whatever Mozart used.
The problem may be particular to Mozart's requiem.
Step one: in Mozart's 'Lacrimosa' the 'i' is sung on two notes.
Step two: In the musical score the word might therefore written under the notes as 'la- cri- i -mo -sa'
Step three: a printer, seeing 'lacriimosa,' would set it as 'lacrijmosa' or 'lacrymosa' with two dots, or plain 'lacrymosa.'