There was a question in the comments from a reader who's picking up med-chem, and I thought it was worth answering out here. I've been meaning to shore up the "Pharma 101" category, and this is a good opportunity. So how, in a case like that compound in the previous post, do you increase a compound's half-life?
The first thing to do is try to figure out why it's so short. That's almost certainly due to the compound being metabolized and excreted - once in a while, you'll find a compound that quietly partitions into some tissue and hides out, but for the most part, a disappearing compound is getting chewed up and spit out. For one that's being injected like this, you'd want to look in the blood for metabolites, and in the urine for those and the parent compound, and try to see how much you can account for. No point in checking feces or the bile contents - if this thing were dosed orally, though, you'd definitely not ignore those possibilities.
Looking for metabolites is something of a black art. There are plenty of standard things to check, like the addition of multiples of 16 (for oxidations). Examination of the structure can give you clues as well. I'd consider what pieces I'd see after cleavage of each of those amide bonds, for example, and look for those (and their oxidation products). The bromine and iodine will help you track things down in the mass spec, for sure. That phenol over on the right-hand side is a candidate for glucuronidation (or some other secondary metabolite), either of the parent or some piece thereof, so you'd want to look for those. Same thing could happen to some of the free acids after cleavage of the amides. And I have no idea what that difluorophosphonate does, but I'd be rooting through the PK literature to find out what such things have done in the past.
If you can establish some major metabolic routes, then you can think about hardening the structure. What if some of those amides are N-methylated, for example? Can you do that without killing the binding? Would putting another atom on the other side of the phenol affect its conjugation? There are all sorts of tricks, mostly involving steric hindrance and/or changing electron density around some hot spot.
Update: a commenter notes that I've left out prodrugs, and that's quite right. A prodrug is a sort of deliberate metabolism. You put in a group that gets slowly cleaved off, liberating the active compound - esters are a favorite strategy of this sort. Much of the time, a prodrug is put on to improve the solubility and/or absorption of a compound (that is, something polar and soluble grafted onto a brick), but they can certainly influence half-life, too.
The other major strategy is formulation. If you really can't shore up your structure, or if that isn't enough, then you can think about some formulation that'll deliver your compound differently. Would some sort of slow-release help? These things are trickier with injectables than they are with oral medications, from what experience I've had, but there are still things that can be done.
So that's a short answer - there are, of course, a lot of details involved, and a lot of tricks that have been developed over the years. But that's one way to start.