Why do they do this?
Well, its because the NPC you made really added to their gameplay experience.
It may be because the relationship hinted at really worked with their concept. Perhaps their aloof elf rogue really gains an extra dynamic when he grudgingly adopts and brings along a 10-year-old human war refugee.
It may be because the NPC is really interesting and suggestive of a whole range of secrets and ideas that the players want to explore. Perhaps the dodgy mechanic seemed to have some sort of perhaps Mafia or government connections and they want to know more about them and their unusual skills.
It could just be because the NPC is funny, lively, or otherwise has a really good and intriguing personality which the players really enjoy. Players will always attempt to get more of the kind of the things in the game that they enjoy. Its only fair. And, in fact, its a good thing. Otherwise, how would you ever be able to adapt the game to their interests?
In the end, whatever the reasoning, the result is the same. An NPC that you may, or may not, have had long lasting plans for is now being treated with greater importance and focus. I suggest going with it. It can only add to the game and you can always use their favourite NPC for plot development rather than relying on the ones they find dull.
Attachment is a powerful glue and one that is nigh impossible to artificially create so when you find them attached to someone -- take advantage of it but be aware that the players will hate it if you find some way to automatically remove them. Kidnap them, sure. Threaten their lives, sure. But if you kill them without giving the players a fair chance to rescue them than they will be really resentful about it. As I said, attachment is a powerful influence and by its very nature it causes negative emotion toward attempts to damage that.
If they die over the course of the game in a fair and honest manner, than it will probably be okay and merely cause a level of sadness and nostalgia. Give the characters (and the players) a moment to grieve their loss and allow them the space to give the NPC the respect they feel they deserve. Different players will require a different amount of time to gain that closure. Some may require a single moment, others might desire a session.
So how about it? Have you ever adopted an NPC or had an NPC adopted in your game?