2003 Mercury 90 hp, 2 stroke. Coming into the ramp yesterday, the alarm buzzer went off. Solid alarm, no beeps. Just replaced water pump and has a strong stream. This alarm sounded on a previous trip is why I replaced water pump. Boat ran fine all day yesterday till headed in. Alarm sounded, stopped the boat, shifted into neutral, checked water stream, shut engine down, turned key on and alarm is solid on, started motor and ran half throttle to ramp, alarm the whole time. Loading boat on trailer and alarm never stopped. It's on when the key is on.
Did a little research and it appears the issue is oil critically low (full tank), oil pump failure, or cooling system. Engine never indicated any loss of power or any other performance issue. Just the alarm..
Where do I look first?

You may wish to consider putting some oil in your gas immediately (like instructed to for break-in for that engine. If you're having an oiling failure, you're adding hard wear to or damaging the motor.

Research the oil level sender. There may be a procedure to bypass that sensor (as a test) to see if the alarm stops. If that stops it, you need a new sensor, which may mean a new tank as I seem to recall that's how you get a new sender. This is a probable failure, but you need to diagnose it unless you are into spending money for no reason.

Add tape to the oil level tank, launch the boat, and carefully mark the oil level with the boat afloat in calm water. Run the boat (with oil added to the fuel to be sure you have lubrication), then check the oil level mark. If it hasn't gone down, you aren't pumping oil.

Consider that your oil system may have gotten air in it. Did you run very low recently, prior to the beep starting? With air, it loses its prime, and won't pump oil. There's a procedure for priming the system.

I checked the oil this morning and it used about an inch out of the tank yesterday. I had filled it and marked where full was at on the side of the tank and every inch from full to empty so I could tell how much it used. Also, turned the key on and the alarm is still on.

That's very good news. That means it's primed, the pump is running, and the engine is getting oil. Over the years, I've seen numerous posts about an issue as you describe, with it resulting in the level sensor being bad. Owners complained they were forced to buy a new tank to get the new sensor - I've never heard that it is sold separately. Isn't that great for parts revenue?

In your shoes, I'd be looking for how to test that the sensor is bad. You should research how that works. Example: If the system design is such that the sensor provides continuity (low ohms) unless oil is low, you can test by disconnecting the sensor cable going to the tank, then putting a jumper across the two pins of the connector on the engine harness side. There are usually such schemes described in service guides, but I don't know how Mercury did it - I haven't had one fail - yet.

If the sensor is supposed to be closed only at low oil, simply unplugging the sensor cable would stop the beep. You should crank it up, let it beep, then unplug the tank sensor and see what happens. If the beep stops, I'd say it's the sensor. I can't see how this test would hurt a thing, but it may mean nothing if the beep keeps going - it could still be the sensor based on how the system is designed.

That's exactly what I thought about doing. If it is the sensor and I can unplug it and stop the alarm, that's perfect. I check the oil every time I get home anyway. Now if I can find that dude. It must be behind the tank and the tank must have to be removed to get to the sensor.

I had a 2003 50hp with the same issue. I don't know if it was the float in the oil tank or the sensor. I unplugged the wire and kept the tank full. I ran it like that until I sold it in 2013 and it's still running like that to this day.

It sounds like the sensor going bad. My 05 125 had the same issue. I could be wrong, but I was thinking the sensor was on the bottom of the tank. Don't hold me to that.

A little more research says you are right. The sensor is on the bottom. Too hot here to work on it today, I'll back it in the shop tonight and see what I can find in the morning while it's cool.

Update: Finally found time to check out the alarm. Turn the key on and the alarm was still active solid tone. Unbolted the oil tank on the motor and pulled the level sensor out and the alarm immediately stopped. Emptied the tank and really couldn't tell if the magnet had fallen off the float or not. I'm going to add enough oil to the gas in the fuel tank to assure the engine gets enough lubrication and run it with one wire disconnected on the sensor. I refilled the oil tank and will see how much it uses on the next trip.If it goes down, I'll be confident it is the switch or the float. If it doesn't go down I'm pretty sure it is the oil pump and will by pass it and mix oil and gas.