You're an aspiring novelist? If so, we have that much in common. In any case, on to answering the question:
'Hold' is a rather vague, general verb. You have an opportunity to give the reader some insight into Mary's emotional state simply by choosing the appropriate verb. Does she caress her finger (fondly) if its to be removed and this is perhaps the last time she'll be able to touch it? It it painful, and so she grips it or her hand clamps over it? As hinted by the last example, a phrasal verb may be appropriate too, if its more revealing and in line with your vision. So you need to ask yourself, 'Why is Mary holding her finger?'
If you decide to use 'hold', I don't think the prepositions you've included are necessary. There's no need for either 'on' or 'onto' in this context. They're only additional words that add bulk to the sentence. Incidentally, all three sentences mean the same thing, as far as I can see.
If there's some significance in Mary's touching her finger (which there should be if you've thought to include it), then maybe you want to have that instance stand out, which would be better achieved by breaking up the sentence.
So: Mary peered down at the beach. She 'held' her finger and sighed, sinking back to her seat.
Having written the above sentence, it seems you could consider bring the 'sigh' closer to the act of 'holding' her finger. The two are connected, and if you include the fact that she sighed, you might not need a more specific word for 'held'.