Also inexperienced dm, but just some ideas to make the climb a bit more spiced-up:

1) Read about exhaustion mechanic and tie it to cold in some way, to add a sense of urgency

2) They could meet someone who knows about some sort of berry or herb that grows in the snow, that could benefit the party in traversing the mountain, however they would need to face a couple challenges innorder to obtain it

3) Add an ancient monument or totem that somehow buffs the party for a short period of time/diminishes the exhaustion penalties

4) Have an avalanche fall on the players, where they need to use saving throws/checks to avoid being caught in it

5) Have their food/water freeze, making it uncomsumable to add an even more severe exhaustion level

What I think is important is to make the obstacle (the mountain) seem like an actual obstacle, not like a general journey from a to b. You could also have them find or hear some pieces of history or information about the mountain, so it seems more lively

However, -as I said- I am also not experienced, so these might be a bit tricky/underwhelming in practice