This reply is late in coming, but I figured someone else might have the same question later on.
To add someone as a friend on My.Lotro.com, you will need to find them on the site first. You can either do a search on their character's name, or their forum name if you know it. The forum (player) name is the best to search for, as that will take you right to the page where you will find a link to request them to be friends.
There is one snag however. If the person you want to add has chosen to remain an Anonymous player on the My.lotro.com page, I don't believe you will be able to add them as a friend.
this is a pet peeve of mine. i think one of the best features is having friends via "my lotro". yet the function is the most counter intuitive design i can think of atm. it's like the devs never tried using what they designed.
March is national target forward healing month! do your part!