This means "Ours by the sea"* (or, if you want to supply a noun where Latin will let an adjective do the job of a noun "Our place by the sea"). Thus, nostrum is nominative and mare ablative. It would be a nice motto for a family with a sea-side home, or for a sea-side city.
Alternative interpretations, with nostrum modifying mare, are way too grandiose for most coats of arms. To call a body of water mare nostrum, you need to surround it entirely. Turkey might refer thus to the Sea of Marmara, but this is the motto of a Brazilian city on the Atlantic, which pretty well excludes that meaning.
Nonetheless, most places on the internet (including Wikipedia) translate this motto as "for our sea" or the like, which just shows that you shouldn't rely on the internet to interpret Latin. People who don't know Latin are probably relying on the familiarity of the phrase mare nostrum from Mussolini's use of it to refer (very) aspirationally to the Mediterranean. The fact that the syntax with pro doesn't work out that way should in this case be an additional clue that you need to reconstrue.
For mare as an ablative, see A&G 76a3.
*Literally "in front of the sea" or "before the sea" but English idiom in this circumstance prefers "by the sea".