The Hebrew word for "elbow" - מרפק marpek is not of biblical origin. It first appears in Rabbinic Hebrew, for example in Mishna Shabbat 10:3. However, the word does derive from a root, רפק, that has one appearance in the Tanakh. Here is Klein's entry for marpek:
From רפק (= to support). cp. Aram. מַרְפְּקָא, Arab. marfiq (= elbow).
And here is what he writes about רפק:
רפק to support, lean.
— Pi. - רִפֵּק MH 1 he supported, upheld; NH 2 he elbowed.
— Pu. - רֻפַּק was supported, was upheld.
— Hith. - הִתְרַפֵּק he leant against, clung to (a hapax legomenon in the Bible, occurring Cant. 8:5). [Arab. rafaqa (= he helped, supported), Ethiop. rafaqa (= he reclined at the table, leaned upon). Base of מַרְפֵּק (= elbow).]
Let's take a look first at the last form of the verb, התרפק hitrapek, since it is the one that appears in the Bible:
מִי זֹאת עֹלָה מִן־הַמִּדְבָּר מִתְרַפֶּקֶת עַל־דּוֹדָהּ...
"Who is she that comes up from the desert, leaning [mitrapeket] upon her beloved?..." (Shir HaShirim 8:5)
This modern translation (New JPS) relies upon the same scholarship that Klein had, and therefore renders mitrapeket as "leaning." The medieval commentaries, such as Rashi and Ibn Ezra quote the Arabic cognate, but give that as proof that it means "to attach." In light of this Artscroll renders the verse "clinging to her Beloved" and the new Koren Tanakh has "entwined with her beloved." I'm not sure where this interpretation of the Arabic came from - perhaps they knew that rafik in Arabic meant friend, which is chaver חבר in Hebrew, and that recalled the root חבר meaning "to attach."
Jastrow writes that in Talmudic Hebrew the hitpael form of the verb meant "to endear one's self." He quotes Bereshit Rabba 45:4, where we find mention of women who were מִתְרַפְּקוֹת עַל בַּעֲלֵיהֶן בְּנוֹיָן - "endearing themselves [mitrapkot] to their husbands through their beauty."
In more recent times, the verb has taken on another set of meanings: "to hug, to cling to; to remember fondly." The first - "to hug" - is perhaps influenced by the approach of the medieval commentators. The latter - "to remember fondly" - I assume was a more creative interpretation of the verse in Shir HaShirim.
Klein also mentions a piel form - ריפק ripek. I've never heard it used today to mean "to support" or "to uphold," but the use "to elbow" does exist, but it's more commonly found today as ממרפק mimarpek. As Avshalom Kor points out here, that's one of the few uses of the root that doesn't have a positive connotation - instead of support, clinging and fond remembrance, to elbow is to rudely push your way into a place.
Returning to the Arabic cognate, we find that rafik provided the name Rafiq, meaning "friend" or "companion." From Arabic, the same word was borrowed into Swahili, where it became rafiki. That name may be familiar from the Disney movie, The Lion King, where it was the name of the mandrill who through magical and spiritual efforts, helped the protagonists. He was their "friend", and as it happened, was always leaning on a walking stick, while bending his elbow.