I think it depends on how competitive of an applicant you are on paper. If you are already competitive and you think you can do well at NYU (or Sinai), I would do an elective at one of these...While you may not match there (read: likely won't), the LOR will look more impressive to some of the other more impressive IMG-friendly programs...(i.e. UAB, Baylor, Boston University).....On the other hand, if you are more of a marginal applicant, and would really like to be in NYC fo residency, I would consider doing the elective as an audition at Einstein, because Einstein is a place you could realistically match to if you impress them there. Honestly I'm not sure how much weight Einstein-Montefiore places on audition rotations, but I think it would help.
In terms of competitiveness for an IMG, Mt. Sinai would be the hardest. Then NYU. Then Montefiore. By numbers alone, Monte has the biggest residency program in IM of the three.
Find out more information about visas first. Specifically what the programs accept and don't accept. Do an away rotation at the hospital that you see yourself going to residency at. Don't waste your time otherwise. It takes a lot of money and effort to do an away at a place unfamiliar to you.
If you can, do the rotation in the field that the PD subspecializes in. So if the PD is Pulm/CC, then do it in that. If the PD is a hospitalist, do a sub-i or some kind of general medicine month. THIS IS MERELY A suggestion. Do your rotation in where you will shine, but it needs to be of mediocre dificulty.
When you start the away, make sure you swing by the PD's office to say hello and introduce yourself. This is a common pitfall of visiting medical students, because they often either never say hi to the PD or come at the end of the rotation. By that point, the PD thinks -- you were here for a month or more, but never came by... and you want to apply for residency? At the end of your away, visit the PD one last time to tell him/her about your experience and why you want to go there for residency.
You need to time the away correctly. I know a lot of 4th years doing their aways in December and January. Wrong time to do it! Best time really is September. This is when ERAS apps (including yours) start flowing in. August is a good month. October and Nov are also fine months, because it increases your chance of interviewing while you are there! Saves money!
Apply to your aways early. Especially those three programs you mentioned. One of my students wanted a specific rotation at Mt. Sinai, and there was a 1 year waiting list. No joke! I had to call myself just to confirm this. Apply as early as you can, when you are certain.
Thanks for the post.