Bea has nailed this, by the brilliantly simple idea of looking in the DRAE. I feel a little diffident about answering, on that account, but I got there independently before seeing her reference, so I feel kind of justified. And I can flesh it out a little.

It is amazing, really, that this definition of "fincar" as "quedar" is still in the DRAE, because it was archaic 500 years ago. "Fincar" ("hincar") in this sense is really a variant of "ficar" ("quedar(se)"), but they are often confused in Old Castilian, and most importantly, Old Asturian used "fincar" in both senses. This is usefully explained in this reference, which I see Apolonia also found.
http://www3.udg.edu/fllff/zifar/fincado.htm

So the meaning of "fincar" is "quedar". But what about the syntax? It looks like an infinitive. How does that fit in here?

This is the second crucial point. "Fincar" here is actually an apocopated third person singular future subjunctive: i..e. short for "fincare". This is a common tendency, as Ralph Penny explains in his magisterial A History of the Spanish Language (p. 156):

"3.7.1.4 Apocope of -e
Latin verb forms whose final syllable contained a front vowel preceded by a dental or alveolar consonant should lose this final vowel in accordance with regular phonological development (2.4.3.2), provided the previous syllable was open (i.e. ended in a vowel). This rule of apocope is indeed frequently applied to the Old Spanish verb, so that we can observe the following medieval forms [...]:
[...]
3rd pers. sing. fut. subj. estovier, tomar".
http://books.google.es/books?id=ZjcrhyQlFa0C&pg=PA215&lpg=PA...

"Fincar" is another of these. So it's "el vuestro fillo legítimo varón que de vos fincar[e]", meaning (uninverted and with modern equivalent present subjunctive): "[...] que quede de vos". And it means literally "who [in future] remains from you", in other words, who is left after you die, who survives you.

In practice, you may choose to say "your surviving legitimate son". Anyway, that's the meaning.

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Note added at 3 hrs (2013-10-11 15:01:18 GMT)
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Just to add another bit of useless information, I'm very interested to see this two-lifetimes grant (the grantee until he dies and then his son until he dies, and then it reverts to the grantor), because I've seen the same thing in the period I'm more familiar with, the seventeenth century. It looks like an old legal institution.

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Note added at 3 hrs (2013-10-11 15:06:05 GMT)
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Rather belatedly, I've read your question more carefully. So the grantee had no sons. The sense of the subjunctive is therefore "your surviving son (should you have one)".

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Note added at 22 hrs (2013-10-12 10:03:20 GMT)
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By the way, just as a point of interest, the future subjunctive (3rd p. sing.) of "fincar" in modern Portuguese is also "fincar".