The prehistory of the Latin lacrima or lachryma is interesting. The Greek is dakru. They are both cognate with proto-Germanic *tagr- which gives 'tear'.

Now the change from D to L in Old Latin has several other sporadic examples; it may have been an influence from some other Italic language.

If the CH and the Y both only occurred in mediaeval Latin, you could write them off as spelling mistakes, since by then CH and C were pronounced the same, as were Y and I. But this is older; there is more going on here.

CH (like PH and TH) is usually a sign of a Greek borrowing. These were Greek sounds and didn't occur in Latin. But they did occasionally show up in native Latin words, such as pulchra 'pretty', generally in a cluster with L or R. It may be the native sound was somewhat aspirated in that position. (Though not universally: we don't see *Chlaudius.)

The alternation between I and U also occurs in a few other words: optimus 'best' appeared as optumus in Old Latin. It derives (IIRC) from a syllabic nasal; and before M it might have been a partly rounded vowel. In this case Y might be an appropriate spelling.

There is another interesting aspect to this word. The -ma is a noun formative, and the L is secondary, so the PIE root for 'tear' was *dakru-. Now it has been suggested that this was earlier *wd-akr-u, with a zero grade form of the root for 'water', and the akr- part means 'sharp'. That is, 'tear' might very anciently a compound meaning 'sharp water'. Against this, the loss of initial W like that is not known in other PIE.