I think what makes me use a unit in the first place is how well they do in the chapter that you get them at. Let's take my favorite character in fiction, Inigo, for example. The paralogue you get him at is painfully difficult for that point in the game (for me, at least), so having a unit with high skill and comes with a Killing Edge, my favorite weapon, so that he's a crit lord automatically appeals to me. Now what SELLS me on a unit is usually either their personality, base class or my personal experience with them. Let's take a unit from that one time I emulated FE7, Sain. Already I like him because he has a similar personality to Inigo (flirty, fellow comrades are done with his crap, always smiling), he was in a well-rounded class that I liked to use (and several others of that class where already my favorites), and, despite having a constant crit rate of 2%, he was still (somehow) a crit lord.
Another thing that makes me love a unit even more is when their looks. I find people, primarily females, with red hair and/or glasses hot (So Miriel and her son are A+'s for me), while I find males with a british/semi-deep voice or mono-colored (black, grey, white) hair attractive (Laslow, Shura, Lon'qu). Anyone with purple hair are automatically used a good deal (Donny, Erk, that mage mercenary dude from FE6, when I'm ever going to play it, also is going to be used a lot).