If a song is a song nothing will make it a hymn...
except...
revised content and focus.

A song may indeed be a religious song...
it may be devotional in nature...
is likely somewhat subjective in focus...
is less likely to have a fundamentally theological bent...

A hymn is focused totally on the deity....
it is not concerned with subjective experience...
it is fundamentally theological in substance and content

Thus -
songs like 'Just as I am without one plea', 'Were you there...', 'We gather together...', 'Here am I, Lord'. and so forth are not hymns. They are about me, we, us, etc., and how we feel. Nearly all the sorts of songs sung as 'gathering hymns' fall into this category and are totally in-apt for liturgy... private or extra-liturgical devotion, perhaps, but not liturgy, which is focused totally upon God. (We will not go into the matter of 'gathering' itself being a concept foreign to the mass, because it focuses on 'us'. We do not 'gather'.... we 'enter into his courts with praise'.... praise of him.)

Thus -
hymns such as 'Immortal, invisible...', 'Praise, my soul...', 'Holy God, we praise thy Name', and so forth are hymns. A hymn, by definition, extols the deity, Who, in the case of Christians, is the Triune God, the One and Only True and Living God.

This is not to suggest that 'songs' are worthless. Quite the contrary. But their proper place is not the God-oriented mass.

Then, there are some which rather straddle the line, such as 'Come down, O love divine', or 'Adoro te', which is replete with theology and adoration, but from a studied subjective perspective.

Of course, this is all about text. The matter of music-text marriage is for another discussion. Suffice it to say that a 'hymn' or a 'song' is the text, as distinct from whatever tune to which it may be sung.

One of the most disappointing things about too many of our Catholic 'hymnals' is the number of songs (not hymns) that focus on trendy social issues and paraphrase scripture in a manner so as to re-inforce them. Of course, there is nothing at all (really, there isn't) with being made aware of these issues, but the mass is not the forum for singing about them. Bible study in the parish hall, maybe. Other forms of devotion and ministry, yes. But the mass is where we worship God and focus entirely on his Triune majesty and the unbelievably sacred eucharistical act. Anything that does not serve this inviolable focus is foreign utterly to the mass.

@ M. Jackson. I am using the term 'song' to indicate a musical piece with a vocal component as opposed to an instrumental musical piece, so all hymns would be songs, but not all songs hymns.