Is there one word which can substitute the meaning of "The desire to feel special to someone"?
Well, there's 'needy':
- (Of a person) needing emotional support; insecure.
He was desperately needy in his sexual and emotional demands.
'Neediness' is fairly close to the meaning you're asking for, in that the person who is 'needy' could need another person to express their love for them. It's a pejorative term BTW.
The concept of 'codependency' might also be relevant:
People with a predisposition to be a codependent enabler often find themselves in relationships where their primary role is that of rescuer, supporter, and confidante. These helper types are often dependent on the other person's poor functioning to satisfy their own emotional needs. Codependency often involves placing a lower priority on one's own needs, while being excessively preoccupied with the needs of others.
Again, this is a pejorative term. Codependency is the desire to feel special to someone by being needed by them, though at a cost to oneself and/or to the person being 'helped'.
Trying to find less pejorative terms, a person who is sad because they are without close friends or companions might simply be 'lonely':
Sad because one has no friends or company:
- Living as she is was miserable and lonely with no friends or loved ones to care about her.
- The new girl's supposed to be lonely and have no friends for at least a month and a half.
- She could not understand how such a great elf with so many wonderful subjects and friends could be so lonely.
The OED defines 'lonely-heart' simply as a "a sentimental name for a friendless person", but I notice that this definition hasn't been fully updated. These days, in the UK at least, 'lonely-heart' carries a very definite connotation that the person is starved of romantic company. In British English, personal advertisements are called 'lonely hearts columns'.
Someone who feels insignificant or unimportant feels that they aren't special. Such a feeling is typically accompanied by the desire to be considered special by someone. These words can therefore be used to imply the feeling you seek in general. For example, "I wish I wasn't so insignificant." can be safely taken to mean "I wish someone would think I'm significant/special."
Another related possibility is the concept of validation. A person seeking validation is looking for another person to validate their existence, i.e. to confirm that their existence isn't meaningless, pointless, or inconsequential in the validation provider's opinion. Often, someone seeking validation is even looking to be praised or appreciated.