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I have recently had contact with an acquaintance from Eastern Africa (I think from Ethiopia or Somalia). When the person greeted me, she shook my hand while holding her elbow of the right arm with the left hand (see image). After this happened a few times, I figured it was not just a coincidence.

Does this gesture have any meaning?

It's a sign of respect. A bit like using both hands to shake yours. (Source: I lived in Africa for something like 7 years.)

I personally found this more common in West Africa than East Africa.

Edit: Perhaps as clarification, the right elbow typically is held with the left hand, as if to support the right arm. But, I've also often seen 'just' touching the right forearm with the fingers of the left hand.

As MastaBaba said, it's a sign of respect. As an Ethiopian, most of us were taught as children to hold our hand (anywhere on the forearm) when greeting elders (basically anyone older). It is considered rude to shake hands without holding one's arm, and in most cases the person (elder) will be offended. Sometimes, the person may also look down during the handshake, which is also another sign of respect.

Having lived in Uganda Africa for almost 8 years I think I can answer your question.

Here in Uganda its a sign of respect. I often place my hand on my elbow when shaking the hands of the elderly because I acknowledge their old age and their knowledge. It means they acknowledge you and respect you. Like @Kidus answered its considered rude to not place your other hand on some part of your forearm. Sometimes the person will nod his/her head toward the ground while shaking your hand, Its also a sign of respect and it would be disrespectful not to follow suit.

I think it is the respectful way of shaking hands. I assume this from reading The No 1 Ladies Detective Agency books (set in Botswana) where the main character refers to this as the proper way of shaking hands. A foreigner who shows respect by shaking hands this way goes up in the character’s estimation.

Just as another place where it is done, this is the standard way of shaking hands in the sport of Taekwondo. See this page for example:

Shaking Hands

A Taekwon-Do handshake is always given with two hands. With the right hand extended, place the left hand palm down under the right elbow. This is a gesture of respect and should be used at all times, even outside the dojang, when giving or receiving items, as well as when shaking hands.

My experience is with Nepal, transaction like paying or gifting are conducted with the right hand. Placing the left hand on the right elbow connotes greater respect or formality to the transaction. Adding the hand to elbow is like changing "Its nice to meet you" to "Its very nice to meet you" or "Thank you" to "Thank you, sir".