Hi,
Is this a AIM screen Name ?
You can get valid AIM names from AIM themselves or an @Mac.com name and MoblieMe names work.
If this is the case then the Buddy will not see anything to indicate you have added them.
However several settings can influence things.
1) If you click on the Status Message (Available or Away) the drop down menu in iChat 4 and 5 lets you set Invisible.
This means your Buddies will not see you On Line.
The same is true the other way round if they are set to Invisible.
2) In the iChat Menu > Preferences > Accounts > Security
Allow All should show up everybody.
Allow Buddy List only will only show you On line to those people you have in your Buddy list (you will not see people who have not added you if they have this setting)
The Block options are similar.
Jabber Lists
These do send a Authorisation Request when you add a Buddy.
Unfortunately iChat only send it once.
If the Buddy is not on line when this happens they do not see it and you have a Buddy that always read "Awaiting Authorisation"
It is best to remove them and arrange a time to be sure they are On line and then add them.
(Alternatively add them in a third Party app like Psi which will keep sending the Request for Authorisation until then either Accept or deny)
A third alternative is to tell them your ID and let them Call you. At this point you can use the Add Buddy option in the Buddies menu that should also list the current Chat person.

10:51 PM Wednesday; June 15, 2011
Please, if posting Logs, do not post any Log info after the line "Binary Images for iChat"
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb( 10.6.7)
Mac OS X (10.6.7),
"Limit the Logs to the Bits above Binary Images." No, Seriously