Okay. Ever since I saw the Mass Effect Andromeda trailer and saw the ship thing, I've always thought it looks like the Citadel because of the whole arm things it has.
So... kind of based on that, I'm thinking/guessing something like this:
You awake to the sound of an alarm. You grab your pistol from your desk drawer, just as a large Reaper passes by your Citadel apartment window and starts destroying things. Your Omni-tool lights up with an encrypted call from the Asari councillor. You accept it. She tells you that Shepard's plan is hopeless. A fool's errand. A waste of resources when the future of organic life is at stake, and that the Ark Protocol had been activated, and the rest of the crew was waiting. You tell her you would be there as soon as possible, and hang up. You fight your way through Reaper forces to the Citadel tower, where the Ark is docked and ready (you might be able to choose to rescue someone or not on your way there and take them with you). You rush on board, but the Reapers start to attack the ship while its leaving. A Sovereign-class ship lands beside you, and you only just manage to escape with your crew. You set a course for Andromeda, then are all put into cryostasis pods while the ship travels.
You wake up a several million years later (not that you know that), with only half of your crew alive due to the ship starting to run out of power. The Reapers had damaged the drive core, so the trip took longer than expected and the AI had to reroute non-critical power to keep you and the remaining crew alive. All you have is a recording of Commander Shepard's voice to keep morale up, and a QEC to contact the council on. But no matter how much you try and contact them, you never get a reply.
You remember your mission - to find a new home for the organic life in the Milky Way. You plan a route to the closest planet, and start to explore.
Benefits of that:
You get squadmates who are "old" aliens (ie you're not forced to pick from the new aliens BioWare have invented for potential squadmates).
You have a link to the old games in terms of story rather than being something completely different.
It adds a feeling of isolation/emotion to the start of the game, and could be a reoccurring theme throughout.
It's not a direct copy of another instalment of the series (see Star Wars: The Force Awakens for an example of that).
The person who you may or may not have rescued could be a potential romance option and cause conflict within the remaining Ark crew members
Disadvantages:
The fact I thought of it.