Why did Voldermort wanted to kill especially Harry?

[fotouba]

No: HPFGUIDX 36224

      I don't know if it is due to my imagination but I think that in 
the first book J.K.R. says that Voldie wanted wanted to kill Harry 
and not his mother. As a matter of fact he offered to her the chance 
to survive (he would spare her life), as long as she wouldn't stop him 
from killing Harry. Doesn't that strike to you as little bit weird? Why 
Harry? What could he possibly have against a little baby? Why was he 
prepaired to spare Lily's life, when she was more dangerous for him 
than Harry ever was till then? Shouldn't that tell us something about 
the <<relationship>> between Lily and Voldermort? I would be 
really glad to hear to whatever anyone has to say about this matter.
After all, I may be wrong:-),
                      starlight.