Hey @raphael oliel,
To answer your question:
1. Refinance interest rates are kind of always higher than what you would get when purchasing a home. It's important you understand how much interest rates affect your monthly payment and consequently, cash flow. Do the numbers before going this route. It would suck to have to fall behind on payments on the home you were raised in.
2. This sounds like your best option to me. You can use your savings to do an FHA loan on a multi-family property. I wish we (my wife and I) had done this looking back even though at the time we thought the price we bought our Airbnb for was high at the time. Hindsight is 20/20 but we made a good decision nonetheless.
3. This sounds like more of a question of comfortability for Mom. This really depends on if she wants to leave what she's grown accustomed to over the span of your life (i imagine) living in one area and relocating to another where ease of travel is a bit different. Seasons are non-existent and friends that have been made may not be as close as they once were. 6% on 250k is better than 6% on 525k (assuming interest rates are the same for purchase & refinance but unlikely).
If you do plan on making the move, I recommend an option 4.
4. Move out and rent your home in Long Island. Go into multi-family and do the short term rental route to accomodate for the higher interest rates in the short term until rates get better (fingers crossed). You can use the rental contract from that long island home to help finance a pretty good multi-family purchase. The more east this multi family is and closer to i95 you are (so east of i95 near the beach), the better in my opinion. I'd shoot for at least 4 units so you can have the flexibility of having your mom and sister in their own space for your sanity. However, you're not going to find cheap multifamilies. Expect to pay at least 1 mil for your property. The entire south eastern coast of Florida is ripe for gentrification and equity build as the WFH culture begins to get more engrained in American work and people move down to Florida.
Don't let the interest rates scare you. As long as you can cover that mortgage payment and make a little more (even $1), it's a win in my book. Long-term equity will stabilize and grow but in the short term we're going to see a bit of fluctuation because of the interest rate increases the Feds are doing.
I wouldn't recommend long term rentals unless you're doing luxury multi-family properties catered to northerners because the average floridian can't afford a $2,500 rent but you can generate that via short term rentals.
