Get good at arpeggios - first one octave, then two (three if you've got a 5-banger). Add the seventh. Get yourself a looping pedal and record short loops of any tunes you feel like you want to solo over. Using mainly those arpeggios (hit the scales for embellishment), solo over the changes (i.e. make sure you change the arpeggio with the 'chord of the moment' on the loop).
Are you a reader? If not, become a reader. If you are, get better at reading

Running through some Bach (cello suites reharmonized for bass guitar? They exist...) will give you ENDLESS ideas for solos. Read melody lines from fake books. Write out melodies you hear (start simple - Mary Had A Little Lamb, Happy Birthday, game show theme tunes, etc), then post 'em in General Instruction for critique. Graduate up to writing more complex melodies that you hear - sung verses from the radio, jazz heads, whatever floats your boat, etc.
Finally, find a teacher. If there isn't a good teacher in your area, find a good teacher who does Skype lessons (I hear Anthony Wellington is an excellent instructor, regardless of whether it's in-person or over Skype).