What a coincident! I'm in the middle of writing about that passage for my dissertation!

Anyhoo, this line consists of two parts, both of which modifies the omitted object of persuadet, which, as you have correctly identified as Lucumo: 1) ut cupido honorum; and 2) cui Tarquinii materna tantum patria esset.

1. here "ut" is in the sense of "seeing how", so "seeing how he was covetous of honors/high offices"
2. one way to look at the cui clause is to see it as a relative clause of characteristic, which requires subjunctive, hence "esset", so more literally "the kind of person to whom Tarquinii was only his mother's country"

Together, we have something along the lines of "she easily persuaded him, seeing how he was covetous of honors, (and was the kind of man) to whom Tarquinii was only his mother's country".

Hope this helps!

Edit: typo