I have a very neurotic and needy cat... should I get her a friend? If so, what species of friend?

Some background! Daisy is 11. She is very needy – if I’d let her, she’d sit on my lap all day (she does when I stay home sick). Some of her neurotic behaviors include leaping onto chairs and hitting me when I walk by, and jumping onto my lap while I’m at the computer and biting my arm (not gently, but never breaking the skin) when I don’t pet her. She is always angriest on days when I work late, so I think she’s just lonely. :(

She used to live with my mom where there are 4 other cats. I took her because she was bullied by a few of the others and seemed unhappy – the vet actually recommended she come live with me.

However now she seems to be in need of more social interaction than I alone can give her, so I was thinking of introducing another pet into the house. I don’t want to make the situation worse, though. Do you think a new animal friend will help, or just make her jealous? I am considering getting another cat/kitten, a rabbit, or perhaps rats for her to watch (I don’t have a TV to leave on for her).