Wow! Got late to the party, but here are my two cents anyways. I have not read the comments above but will later on.
I will start by saying that it doesn’t seem that big of a deal to me unless she is having problems getting her meaning across, if not, then people would think it is just her accent.
It is not clear to me what her struggle is other than pronunciation. Does she not know when to use one or the other? Does she also struggle with these two in written form?
You started out with:
My mom has asked me quite often: What is the difference between “being” and “having been”?
At least, she does acknowledge that these are two different words in spite of her inability to perceive the difference between the two sounds.
From what you mentioned, in this case, I would not take the grammatical route as such. Instead, let’s take a problem-solving approach and break down the problem in different parts to find realistic solutions. First, let’s identify the issues and then provide a few practical solutions so you can help your mom. This needs to be done in steps so that you can go gradually.
Issues:
1. Phonetics: She cannot perceive the phonetic difference between these two verb forms.
2. Usage: It is not clear to her what situations require using one form or the other.
Possible solutions:
Phonetics:
1. Google being pronunciation and been pronunciation, you will get a speaker icon and a visual of lip movement for each word.
2. Click the speaker and have her listen to the two words a few times.
3. Stress the significance of lip movement and how important it is to mimic that. (Sound differences are the result of lip movement and tongue positioning.)
4. Then, ask her to isolate the sound of the endings only and repeat a few times.
5. Find other words that have the same endings and have her repeat them a few times.
6. Make a few flash cards for her with all these words using the same colors that you had initially assigned for each verb form (see below). Repetition is the key.
Usage:
1. Get two sheets of paper and two different color markers. Write each verb phrase on a separate sheet using a separate color for each: is/are being and has/have been
2. Give her a few examples in Spanish first (that way she will have a more precise point of reference), such as:
• Being: El español está siendo traducido incorrectamente en EE. UU. that is Spanish is being translated incorrectly in the US. (This is something that is happening right now [currently].)
• Been: El español ha sido traducido incorrectamente en EE. UU. desde el principio. that is Spanish has been translated incorrectly in the US from the get-go. (This is something that started out long time ago and it is still happening in the present time.)
3. Explain to her that these two situations refer to the same thing but are taking place in two different points in time.
4. The key element to understand is when these actions are taking place.
Needles to say, you need to understand the difference of the two forms very well (both phonetics and usage) so that you can explain them to her. Arm yourself with a variety of examples where you can discuss the different implications of using one form or the other. The practical approach is better in these cases. Do not overload her with too much grammar detail that won’t help her at this point (she is not a student).
PS: Let me know if you need help understanding the difference and use of these two verb forms (present continuous vs present perfect) because I approached the issue assuming you did understand the difference but did not know how to explain them to your mom.