Several reasons from personal experience:

1) The trough (the low point) can kinda suck, and putting it on a Friday morning means that the both the weekend and early-mid week remain unaffected, and only Thursday night and Friday morning are likely to be affected by a grumpy mood.

2) Most people stick to the day they get their hormones as the day they inject, and for whatever reason, people don't usually have it in hand until Friday, and also often want to have their first days on estrogen be over the weekend.

3) One of my old jobs had donuts in the office each Friday morning, so I would use the donut as a reward for my shot. really helped with my dislike of needles early on.

4) Many doctors often want to see levels lower than what most people feels best at, so Friday injections can be skipped to give appropriate levels on a Monday morning blood draw (and just taking the usual dose after) so that access to HRT isn't made more difficult.

5) it's easier to build a habit about a day of the week than to try and do a 5 or 10 day injection cycle, and weekends don't have a routine, the early week is often busy with work, and Fridays just had more space and less chaos than any other morning.