Is there a term for embedding an acronym or initialism inside a friendlier-sounding name for a desired effect?

At my job I have created an application which supplements the functions of the main system the employees use during the day. The main system is referred to with a Three Letter Abbreviation (TLA).

Because I wanted to be clever and memorable, I gave my application a friendly-sounding name (in theory at least, to native English speakers) which I created by adding vowels to the existing abbreviation. So the helper application for the TLA system is named "TeiLA" (not the actual letters or name, but the idea is the same).

This has created the desired result in that users say things like "Have you asked TeiLA for that information?" and even "TeiLA is my best friend!" I've been asked to talk about the work I've done and specifically to touch on name selection, except I don't know what to label what I did.

Is there a word or concise phrase which describes that action or process of transforming the letters into another name? Or, along the same lines, a word or concise phrase to describe the intent of making the name sound friendly or approachable to its users?

You could call the way you have treated this acronym personification.

Merriam-Webster defines personification as:

attribution of personal qualities; especially, representation of a thing or abstraction as a person or by the human form

In your case, you have attributed the acronym with a human sounding name and a personal role in your office.

You have taken the initials of the application and encouraged people to use it as a pronounced acronym, perhaps by adding the extra letters in office communications about the application.

Whether initializations that are not pronounced are or are not acronyms is a matter of definition, but in general use most people would consider FBI an acronym, although the letters are pronounced individually. Any U.S.-based list of acronyms will include FBI and similar initializations. Whether a or an is used to precede such an acronym depends on whether it is pronounced as letters or as a word, with the indefinite article being chosen accordingly.
Consider the Screen Actors Guild. The elegant short-hand way to refer to this organization is "the Guild," but of course there are other guilds. It is abbreviated SAG. The Guild itself objects to pronouncing this as a word and would not use the phrase "a SAG meeting." Instead, they'd say "an SAG meeting," where SAG is pronounced ess-eh-gee.

I'll leave this here as a starting point since it's the closest thing I've been able to think of: familiarization.

As in, "TeiLA" is a familiarization of the name for the TLA widget system."