The best source of info for the area is the relevant OS map, and HCC's online rights of way map, here:
http://localviewmaps.hants.gov.uk/LocalViewmaps/Sites/ROWOnline/#. You're looking for the brown lines to denote BOATs (Byeways Open to All Traffic)
Wiltshire's effort is here:
http://www.wiltshire.gov.uk/communityandliving/rightsofway/publicrightsofwaymapping.htm. Again, you're looking for the brown lines to frive.
Unfortunately HCC is somewhat anti-4x4 and the coast is quite built up, so you have to get out a few miles from Eastleigh to find anything.
There is a belt of lanes which isn't too far away, running from Winchester across to the Stockbridge area, and some are quite good if a little scratchy in places. The last run I did in the area started just north of Romsey, moving across towards Winchester and the A34, then up towards Basingstoke. There are mroe to be had the other side of the M3 round the Alton / Farnham area too, although I've not done those in a while.
Depends what sort of a day you want I supppose - I'll happily dig out my notes for my last run, just shout if I can be of help.