What raised "area"? Where? How long and wide?

You have two options that I can picture: Jackhammer it out and pour new level with the other. Or build a frame with floor framing sleepers on the lower section to bring the subfloor up to the height of the raised part. That will be easiest, but I may not have a clear idea of what you really mean.

Typically, garage floors have a slope toward the door(s) and a step between the home and garage. You'll need to consider both when you assess the cost and feasibility of converting your garage to living area. You'll also need to deal with the consequences of whatever fuels and chemicals may have been absorbed into the concrete slab.

In order to be liveable you need to meet HUD standards of insulation, wiring, heating, cooling, egress, etc. I'm not sure a garage built in 1959 could even be upgraded. What's your budget? What will you do with your cars?

I should have been more clear... the house was built in 1959, but has had several upgrades during the ensuing 60 years. The two car garage that is attached to the house had been converted to a room before we ever bought the house (20+ years ago.) Originally the driveway came from the east (front of house) , but was moved some time in the past to the north. We have never used the space for our cars.

During some remodeling in the past, the garage doors on the north were removed and a large window installed, as well as a door out to the driveway area. Two windows were put in on the east. There is a door to the west that has a big step up to the patio and deck. A wall AC/heating unit was installed. There is no direct entry into the house. We have primarily used this space as storage that we are now clearing out. Currently, I don't have a TV room or den, or any place that will comfortably allow family and grandchildren to gather, other than outside on the deck, which is not very comfortable in Texas summers or during the winter.

The room itself is about 19'X 24', with the raised area, 15' X 4'4" on the west side. My plan, after talking with contractor, is to extend the raised area another almost 10" into the room to give more space for a some storage cabinets and a work counter, and another 4' so it meets the south wall. That will give me a raised area of about 19' X 5'2", roughly.

I'm getting the appropriate permits and using licensed contractors, so I'm not trying to cobble something that will be unlivable. I plan to stay in this house another 30 years at least. However, having this space will bring me great joy, as I really don't have an entertainment area for my grandchildren at this time.

My biggest question is a design one about the flooring. Since I have that raised area, what would be a good way to transition the floor? I would like LVP, but wonder about that lip from the raised area to the "main' floor. It actually is about 3". Rather than LVP on the raised area, would another flooring be better? The previous owners had put down an indoor/outdoor carpet which wrapped down over the edge. I don't love it.

My current budget to get permits for electric, remove the fluorescent lights and install additional lighting, remove the indoor/outdoor carpeting, replace the door and framing on the driveway side, extend the raised area and install the flooring is about $9000.

Pictures say 1,000 words, and money talks. I don't think $9K is going to get much of a conversation going about conversion.

If it was meant for cars, chances are that the city will approve leveling the area. You just need a contractor that can do that for you. Since most city codes are getting updated regularly, you need to submit new plans to see if you can level that area and make it so that the transition is more smooth.

If you must keep it, it can be like wood or carpet flooring on your staircase that transition into wood/tile floors.