A few things..
The not making eye contact.. Is that even with you and your family members or is it every person? If it is also with you and your immediate family.. I would have her tested for Asperger's Syndrome.
Here is a link http://www.webmd.com/brain/autism/mental-health-aspergers...
Otherwise
Is she bright?
Is she super mature?
Does she have any special interest?
What does she enjoy doing?
Our daughter has always been shy. Very studious.. Enjoys being around adults.
Her interest could be very solitary. Reading, Art, music..
She had a few friends in HS, but they were all kind of Nerdy/Geeky like her.
She joined the Art Society, Was a member of the National Honor Society and so this lead to lots of Volunteer Opportunities.
As parents we all had to kind of prod them to get together and do things.. Especially in the summer.
We the parents facilitated by allowing day long Geeky movie parties.. we had the food and snacks.. Or taking them to some of the festivals and making them go and "Have some fun for gosh sakes". They liked hanging out at one of the book stores here in town with a cafe..
Gather them together and meet up at a pool.. Or an out door movie event.
I also made sure our daughter volunteered in summer programs so each day she had a place to go.. This was great because she got many babysitting and tutoring gigs out of this.. The young kids loved her.. she got along great with the teachers.. so the parents totally called on her.
The happiest she has ever been is once she got to college. She completely bloomed! She was surrounded by a whole campus of people like her. Mature, geeky braniacs.. And then the more outgoing people dragged her around with them.. so she did become social.. I almost fainted when she told me she had been elected to be the House Vice President!!!!!