No way to know until you start pulling some tiles down- If I were you I would pull down all the tiles NOW (leave the grid and lights) to see what you are working with. If the framing looks good paint can be cool, if there are not a lot of things hanging below the ceiling sheetrock is the best answer, framing a whole new ceiling would be expensive. Even if you keep the tiles led can lights will give you much better lighting than what you have now

I agree, more lighting would really improve that space. The other option is to remove the drop ceiling and paint the exposed beams. If you search houzz photos for "exposed basement ceiling" you'll see lots of examples.

Replacing the tiles may be a good option, there are some very attractive tiles on the market. You may be able to get cross braces if you'd like to use square tiles. I can't tell if the existing drop ceiling is level or has any sagging.

The drywall ceiling is a good option, it will give the room additional height. You could remove the existing ceiling yourself, then have an electrician install lights. The drywall doesn't have to happen immediately, it can wait until you have the budget.

Sometimes you just have to bite the bullet. Do the drywall and add recessed LED lights. Will look a thousand times better than it does now so the expenditure will be worth it. Will be a nice hangout space for your kids as they and their friends get older and move from LEGO building to playing ping pong or video games.

For most of an existing basement we exposed the existing ceiling joists and had them painted. Also stripped the existing concrete slab and did acid staining. I wouldn’t say it was an easy project but wanted to avoid drywalling the ceiling.

my guess; there is insulation under the drop ceiling so I think having exposed won't be an option.

What Hallett said, pop out the old ceiling tiles now and see what mysteries you have in that ceiling.

It’s possible that former owners dropped the ceiling lower than necessary on purpose to cover up utilities and beams, and sacrificing height there that you need now.

Drywall is not great for a basement ceiling if you plan on making changes to the upstairs, or if the ceiling is full of shut off valves and transformers and plumbing access and ductwork. You really don’t want to entomb the main floor.

If you pop out the acoustical tiles and find a mess of old wires too unsightly to be seen, instead of covering it up you want to get an electrician to sort through and clean it all up anyway. An electrician should sort through what wires are allowed where per code as far as what wires should be in conduit.

There are quite a few nice ceiling tiles out now. Armstrong makes some. Also, maybe tongue and groove depending on what you like

I have a similar situation but am in no rush to upgrade the basement because, unlike you I don't need the space right now. One thing to consider, though . . . I remodeled my kitchen, bathroom and repiped the house. It was SO much easier (and cheaper) for the trades to access the plumbing and wiring through the basement. They just popped out a few ceiling tiles as needed and no expensive floor or ceiling repairs were needed. So my advice . . . think ahead.