Note the ~ on the front of the version. That says that the version installed must be at least 0.4.3 but must be less that 0.5.0, so it won't let you install 0.6.0. It should, I believe, say
"node-red-node-sqlite": "^0.4.3",
Which means at least 0.4.3 but less that 1.0.0, so it would allow updates to the current (0.6.0). I am currently investigating gets set to ~. There may be an issue in node-red because I see that if I install via the palette manager it ends up as ~ but if I install via the command line it is ^. Also I find that if I edit the file and change it to ^ and restart node-red it still won't let me upgrade via the palette manager, which I don't understand at all, unless a bug has crept into node-red.
In the meantime the best solution is to remove it via the command line and install it again.
npm remove node-red-node-sqlite
npm install node-red-node-sqlite
which should get you 0.6.0
I also note you are using a beta version of the midnight red theme, so there could be an issue there.
I will start another thread discussing why the versioning issue happening in the first place.
See Npm, versioning and palette manager confusions
