I like both - but especially love
Ada. It's so delicate, classic and beautiful. It works well as a nickname or alone. And, of course, there's the
Ada Lovelace association.
Ida is a little tougher which I like. I wouldn't use it myself though.
"You can have freedom, or you can have peace. But not both at once."
http://www.behindthename.com/pnl/151927