Cessna 150 pilot reported an engine malfunction while in cruise flight over the ocean between islands. The pilot diverted to an alternate non-towered airport and landed safely.

Narrative

Aircraft Component Partial Failure: EngineI was flying home to ZZZ3 from ZZZ2 with my mother on board my personal Cessna 150; we were returning from a weekend in ZZZ2. I am currently training for my instrument rating; but am not yet instrument rated. I had requested and received VFR flight following on switching to departure control on climb-out from ZZZ2. I always get flight following for over-water flights between the islands. At first I was north of; and paralleling; V2 airway; as per ATC instructions to remain north of V2. I then requested to alter course to pass south of the island (south of V2) to avoid clouds in my path; and ATC approved.Approximately 18 miles west of ZZZ VORTAC; I noticed my engine starting to run rough and my exhaust gas temperature rising. I was running at cruise power setting of 2300 RPM; at 80 KIAS; leaned out as normal for my VFR cruising altitude of 5;500 MSL. Everything had seemed normal up until this point. When I inspected the fuel before departure from ZZZ2 I had found no moisture or debris; I had full fuel tanks and full oil; the air filter had looked clear; and my run-up was normal.First I tried enriching the mixture. This helped a little; so I put the mixture full rich. This helped but did not fix the problem. I pulled the carburetor heat; which helped; but the engine continued to run rough. I radioed ATC to alert them to the situation and ask whether ZZZ or ZZZ1 was the closer airport (I do not have GPS or DME). They were both equal distance from my location; so I chose to head for ZZZ since I had landed there before and I was familiar with local conditions there; whereas I had never landed at ZZZ1. I informed ATC of this decision; but I did not outright [Request Priority Handling]. However; ATC implicitly did so for me; asking me to report fuel and souls on board; which I did: 2 souls and 3 hours of fuel.I ran my checklist; but nothing resolved the problem. The engine was surging and losing power; I increased to full throttle and at first I was able to climb at 50 FPM at Vy. At full throttle the engine was initially producing 2100-2200 RPM. It normally produces just over 2500 RPM at full throttle; as it had on takeoff and climb-out at ZZZ2.My goal at this point was to get to within glide distance of ZZZ. I wanted to gain; or at least maintain; altitude as long as possible so that when/if the engine quit entirely; I would have a chance of making it to the runway; or at least to the island rather than ditching in the ocean. Having my elderly mother as my passenger added to my concern about avoiding a water landing. The seas were rough; heavily white-capped with big swells; and I knew that the airplane would likely flip over on landing in the ocean; making it even harder to help my mother out of the plane. We were wearing life-jackets and I had water safety gear - personal locator beacon; lights; signaling mirror; flares; water dye. However; I know from experience how hard it is to find a person adrift in the open ocean; even with GPS coordinates from their personal locator beacon. I am an experienced open ocean sailor; having sailed 60;000 nautical miles in all the world's oceans on very small vessels; and I estimated the wind speeds to be 25 knots and the seas averaging 10 feet.The island has 1000-foot cliffs on the western side by the airport; I knew I had to be able to glide clear of these; and my engine was continuing to lose power. Because of these concerns; I entered IMC in the clouds that extended out from the island: the engine was surging badly enough that I believed it could quit at any moment and I would not have made it to the airport had I descended beneath the clouds. ATC informed me that the cloud bases were 2000 feet; which looked accurate and which was correct. By this point the engine was making only 1800 RPM at full throttle and I could no longer hold altitude. However; I got an updraft in the cloud and managed to get within glide range of ZZZ. I then managedto find a hole in the clouds above the VOR. I made a spiraling descent in VMC through the hole; during which I could see both the VOR and the approach end of Runway XX. Once I was under the clouds and could see the whole airport; I was to the left of the airport and so I made a (nonstandard) right pattern to Runway XY (the correct runway for the winds). I landed without incident; and even though the engine was still running rough; I was able to taxi clear of the runway. I remained in contact with ATC throughout the flight and alerted them once I had landed safely.Although I know that entering IMC could have made my flight worse by increasing task load and potentially causing disorientation; I believed it to be a better option than descending to below 2000 feet outside of glide range to the island. I would have had to descend when I was still well over the ocean in order to get under the clouds. Given that the engine seemed on the verge of complete failure; I chose to hold my altitude as long as possible instead. While I'm not yet instrument rated; I was at least somewhat current on attitude instrument flying; having had two simulated instrument training flights in the previous 7 days; one with a safety pilot (a lapsed CFII) and one with my CFII. However; I feel very lucky that the clouds were not a solid layer and that I found a hole with clear sky all the way to the ground. I was fortunate that I was not in the clouds for more than about 3 minutes.A cargo pilot on the ground at ZZZ was also in contact with ATC during this flight; informing us of cloud ceiling and where the clouds were fewer and thinner. ATC was extremely helpful in staying in communication with me throughout the event; and in ensuring there was no conflicting traffic in the vicinity of ZZZ. Although I knew that a good outcome was up to me; the attention and competence of ATC and the other pilot was helpful and reassuring.Unfortunately; due to the lack of A&P mechanics on the remote island; no one has yet determined the cause of the engine roughness and power loss. Mechanics I spoke with on the phone suggested an induction leak; fuel flow problem; extremely bad carburetor ice; and loss of compression in a cylinder; but no inspection has yet been made. I sumped the fuel and checked the oil and the air filter after shutting down and everything was normal. Oil temperature and pressure had remained in the green throughout the emergency. No one was injured; and there was no damage to the aircraft that I know of; pending a mechanic's inspection.I would most likely behave in the same way again in the exact same set of circumstances. However; I would explicitly [Request Priority Handling]. I have also now ordered a life raft and will be carrying that on future over-water flights in addition to the rest of my safety equipment. I will also continue my instrument training; and once I have my rating I will file IFR flight plans for inter-island flights. I will also make sure to stay proficient in instrument flying.