Hoarders: Abnegation Edition
In the epilogue to Allegiant Tobias returns to his old bedroom in Marcus's house, where he finds a little blue sculpture that meant a lot to him and gave him comfort when he was still living in fear of his father. In the first story of Four, "The Transfer," we see why these objects were so important.
They're a minor form of rebellion for Four. Abnegation members aren't supposed to have sentimental objects, "objects that others would call useless" (1.94) like glasses without lenses (hipsters would never call those useless, hey-o!), discarded computer parts, spark plugs, or a broken bottle.
Four calls them "secret things, and valuable ones, if only because they were so overlooked" (1.94). Tobias, before becoming Four (be-four, we'll call it), is also often overlooked, making it understandable for him to become so attached to overlooked objects. Secret objects: they're just like us.