In Arabic it is said رَسَتِ ٱلسَّفِينَةُ the ship became anchored, and so the verb رَسَا(rasā) reflects a meaning of something or someone becoming steadfastly fixed in its place.
As for rendering رَوَاسِى (rawāsī) into English as 'stabilisers,' although I can appreciate the link to the original meaning, it is slightly incorrect in my view. It would be more accurately rendered as firmly grounded [mountains] with 'mountains' in parentheses because it is implicit to the participle, not explicit.
Alternatively, if you really wanted to use 'stable' then one should render رَوَاسِى(rawāsī) into English as stable [mountains] because the verb رَسَا (rasā) is intransitive, that is, it cannot take an object, therefore, it can be rendered to be stable, but not to stabilise.
The verb مَادَ (māda) means to shake and tremble, and تَمِيدُ (tamīdu) is the present tense form of the verb for a feminine subject.
And so translating وألقى فى الأرض رواسى أن تميد بهم we have And He cast into the earth solidly grounded [mountains] that it should shake and tremble with them.