I have this same issue, @Just_Chevonne - I have only seen the option to "offer to make over" once, and it was while interacting with my sim's boyfriend. By the time I got the option to show up, they were both super tired, and kept trying to go to bed. I don't have control over him yet, and she queued the interaction but then reset and tried to go to bed. I couldn't get the option to come up again, and both of them were adamant that it was time to sleep.
I read Carl's guide and several of the posts at the Sim's forum, but none of them actually say HOW to do this. Of the links that @SheriGR listed, only the sneak peek actually lists anything specific about giving a makeover (at 9:29). At least you can see the interaction, so it answered a couple of questions I had. That sim didn't have much of a relationship with the stylist sim, so it doesn't appear you have to be more than an acquaintance. I also thought maybe it had something to do with either of the sim's moods, but doesn't appear that is an issue (at least before the career actually came out).
I've spent 3 sim-days attempting to get this interaction to show up for anyone - I've only seen it that one time. I traveled to the museum, tried meeting new sims there and interacting with boyfriend there and at home; also tried with household sims with high relationships, using the career-specific interactions and not, waiting for pleasant conversations, etc - nothing works! My game got very "stuttery" and several interactions stalled, with time passing and nothing happening. I cleared the cache and repaired the game, that helped the stuttering but did nothing for getting the "makeover sim" option to appear. I think I've concluded that you don't want to ask them to model their look; it takes forever, seems to get corrupted and stuck in that interaction.
So, although the quick answer is - click on the Friendly interactions and choose "Offer to Make Over Sim", I have not been able to successfully do this. I'm off to try again; it's very frustrating, my Sim has lost job performance and money because she isn't getting paid. It keeps queuing this task for work at home, and I can't get it to work.
I don't have GF; I just have base game; no mods or cc. I'm off to try again. If I find something that seems to work, I'll let you know. Please ping back with any successes you have.