My 11 year old daughter has a tender lump beside her right nipple, not under the nipple. Is this the beginning of thelarche?

Congrats! Puberty. It certianly sounds like pubertal changes breast tisuue can begin anywhere in the breast area during puberty - it does not have to start right under the nipple. Tender is good too, although she may not think so. It's the non-tender breast masses that would be of more concern. And usually one side starts before the other (usually the dominant hand), so the fact it's just one side is ok to start.