I would actually recommend getting another bunny to bond with yours as bunnies crave to be around other bunnies and the companionship they provide.  If you watch their behavior, they groom and love on one another it is quite something to see and to understand that they really do need each other.  Your bunny might just be missing this type of interaction with its own kind or it could just be her personality for now.  At 4 months, she is still just very young yet and has yet to bloom into her full-blown personality.  The time you spend with her now is great for bonding and warming her up to you as well.  Do you offer her treats while you are on the floor with her and play with her or just sit?  You may want to try a little more interaction than petting so she knows that humans aren’t so "boring."  As for the picking up, it takes a little time before they warm up to that idea.  Start slow and make her feel secure, tpick her up then put her right down again once every day.  It sounds like she has really warmed up to you already, give her time and you’ll both grow together.  

i have two bonded rabbits- one that tollerates being picked up and the other that won’t. It’s just a personality thing. I wouldn’t worry about it, if your rabbit jumps into your lap- that’s a good thing! Besides- she is still young, give her time- just don’t be really pushy about picking her up. She’ll end up associating it with a negative feeling.

i would wait until after she’s spayed to have her pick a friend. i think bonded bunnies are generally happier, since they’re such social creatures. most rabbits don’t much like being picked up, so getting her a friend isn’t likely to change that. but an outgoing companion bun can help bring a shy bunny out of her shell.

you might also try holding her and then giving her a reward like a treat. of my three buns only one seems to not mind it at all.

Most bunnies don’t actually like being picked up, though many will tolerate it, which is great when you need to clip nails etc. Just because a rabbit doesn’t fight or squirm to get down, doesn’t mean that they enjoy it. A fast heart rate and/or wide eyes can reveal that they are not comfortable, however, some bunnies absolutely do enjoy being picked up, but it is rare, and your cousin got luck if that is case. The fact that your bunny jumps on your lap to be pet is wonderful!! That’s also not all that common, so count your lucky stars for that sweet gift.

Your cousin also may get lucky with her bunnies being bonded, but her rabbits are reaching sexual maturity, and it would not be unusual if they got aggressive with each other over the next few months. So tell your cousin to keep look out for that as they can really do damage to each other.

I also believe that waiting until your bunny is spayed is best. Especially because of the age she is reaching, even if she’s not aggressive, the scent she’ll be putting out right now can bring out territorial behaviors(aggressive and/or messy) even in another neutered rabbit.