Look, sometimes boring is good.
And in this case, the very boring Horatio is good in many respects. First of all, in a play where nearly every character has some kind of dark side (or at least a few questionable actions on their rap sheet), Horatio exists just to remind us that the otherwise bleak world of The Spanish Tragedy is worth saving. So, he's not only a character of good intent, but he's good for the play—and our spirits. Otherwise, we could easily wash our hands of the whole lot.
For example, hardly any one is nice to the heroine, Bel-Imperia. But Horatio thinks of her needs first when their love begins to bloom: "Thus in the midst of love's fair blandishments/ Why show you sign of inward languishments?" (2.2.5-6). His concern for her inner sorrow may not seem like much, but at least he notices and cares, unlike, well, everyone else. The rest of the men use Bel-Imperia for their own selfish needs.
But if you're waiting for an awesomely meaningful or memorable quote from him, keep waiting. Dude's not exactly Prince Hamlet, nor was meant to be.
Despite being a nicely vanilla character, Horatio ranks relatively high in importance in the character list because he represents an key idea throughout the play. See, it's the memory of Horatio that moves Hieronimo to the supreme heights of philosophical thought and the dark depths of violence and sorrow. Without Horatio's unblotted character, Hieronimo's pain would lack emotional urgency.
And who else could say something like "Fortune is our friend' in a play that rips people to shreds for almost nothing (2.4.16)? It's a genuine Ned Flanders kind of positivity, but given what goes down in the play his comment comes off as darkly ironic to the audience.
Hieronimo carries a handkerchief smeared with Horatio's blood to constantly remind him to kill his enemies. And Hieronimo refuses to bury him. (Morbid much?) So even though Horatio croaks by the middle of the second act, he's always around, sort of. This makes him kind of like the Ghost of Andrea—dead, but present.
In the short time he's alive, he acts valiantly on the battlefield, stands up to the dishonest dealings of Lorenzo, and serves as a protector for Bel-Imperia, who is easily the most abused character in the play. And before you go thinking that he was doing anything wrong by dating his dead friend's ex, know that the dead Andrea fully approves of their match. How do we know? Oh yeah, he's a ghost and he's still around.