i started giving my daughter boiled eggs just after she was a year old, and had no problems with her... but she doesnt have it every day, usually twice a week. hope this helps :)
I gave my daughter soft-boiled eggs not long after she turned one as well. Mind you, she doesn't like them anyway but she was fine with the odd few she had. Its safe for children to have one egg per day.
My wife started giving our boy boiled egg at about 1 too. Another great one they love is a small (with 1 egg and full fat milk) omlette for some variation. We now put cheese, ham and tomatoes onto it and lightly grill it for a minute or two, goes down great... Enjoy
eggs can actually be given if well cooked, hardboiled and chopped or scrambled from 7 months, so really when the baby is ready from this date to eat the texture, probably mushed up scrambled egg would come before hardboiled for ease of eating.