The other answers I see here describe how to tell the person that you would like to let them have your seat, but that doesn't appear to me to be what the question asked.
To answer your question by finishing your example sentence, I would say:
I gave up my seat to an elderly lady.
An alternative form would be:
I let an elderly lady have my seat.
This could imply that she asked for it, and you then let her have it, as opposed to the first phrasing which implies that you did it without prompting.
That being said, one of the comments on this page indicates that a listener may interpret those two phrasings with the reversed meaning, with the second phrasing voluntary and the first forced.
Either way, the two phrasings are essentially identical. Use whichever you prefer.
As far as offering your seat, I normally wouldn't even say anything. I would just stand and move aside when someone who clearly needs the seat more gets on. This is partially because most buses I've been on have signage saying to give up your seats for the elderly or handicapped.