In this sentence the correct pronoun is "mně" (Tak tedy patří mně, neboť já jsem na ně myslel první.) It is the 3rd grammatical case (dative). If you used the other, it would be wrong. Both "mě" and "mně" are forms of pronoun "já", they only differ in the case.
Have a look at this table: http://e-skola.sweb.cz/sklonovani_zajmen.html
There's absolutely no difference in pronunciation between the two.
The translation would be something like "So they(it) belong(s) to me, because..." = "Tak tedy patří mně, neboť..."
February 8, 2016
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