ichpat

The Hebrew verb ichpat איכפת is strange. (It is sometimes pronounced in Modern Hebrew as echpat, perhaps because it is more commonly written as אכפת - without a yod - and therefore looks like another similarly structured word אפשר, pronounced efshar. To hear the word in Hebrew, along with many examples of current usage, listen to this episode of the great podcast Streetwise Hebrew.)

While commonly translated as "to care", I think a better translation would be "to matter" or "to concern", since it is always followed by the preposition "to" as in lo ichpat li or ma ichpat lecha, which mean "[it] doesn't matter to me" and "what [does it] concern you". It is the root of the word ichpatiut - איכפתיות - "empathy" (discussed at length here).

What is the etymology of the word?

It first appears in post-biblical Hebrew, and Even-Shoshan notes that it was borrowed from Aramaic (for example in the Targum to Divrei Yamim I 21:13), where it is a form of the (related) Hebrew roots אכף or כפה, meaning "force, compel".  This is also a theory presented by Klein:

אִכְפַּת intr. v. PBH to pressure, to care, concern. [Of uncertain etymology. Perles connects it with Syr. אֱכַף (= he had regard to, was solicitous, took care of). See אכף ᴵ.] 

His entry for אכף is as follows:

אכף ᴵ to press force.
    — Qal - אָכַף he pressed, urged (in the Bible, a hapax legomenon occurring Pr. 16:26). [JAram. אֲכַף, Syr. אֱכַף (= he pressed, pressed hard, urged), Akka. ukkupu (= to urge).]

In Modern Hebrew אכף means "to enforce",  and akifa אכיפה means "enforcement." Klein suggests that ukaf אוכף - "saddle", also may derive from this root. Jastrow makes the same connection, and offers a common meaning - "burden".

A different theory connecting ichpat and ukaf is presented by Horowitz (p. 90). He writes that

the basic thought here is "resting upon." The saddle rests upon the horse. Ma ichpat li מה איכפת לי really means how does this rest upon me, and figuratively, of course, how does this concern me.

This is similar to the position of the Arukh, who says the root means "to bind" (so possibly deriving from the root כפת - "to bind"), and in the same way a saddle is bound to a horse, this "thing" is now connected to me.

In the footnotes of the Ben Yehuda dictionary, all of these suggestions are discussed, and in the end, none appear convincing. But ma ichpat li? It was fun looking into them!