My interpretation is this is just another example of the Grandmaster's eccentricity. It reminds me of a dialogue in the novel The Light Fantastic:
"Sort of mad. But mad with lots of money."
"Ah, then he can't be mad. I've been around; if a man hash lotsh of
money he'sh just ecshentric."
The wealthy and powerful are surrounded by people who admire them, fear them, or want a piece of the action. That means people like the Grandmaster are surrounded by people who either suck up to him, or cut him a lot of slack - this means he can march to the beat of his own drum, and blatantly ignore social conventions (thus making him seem eccentric), and do things such as not finishing sentences to the satisfaction of others. The humour is in both the deliverer's mannerisms, and the recipient's reactions (Loki seemed accommodating yet exasperated - like a parent of a tantrum monster, Thor seemed equal parts perplexed and distressed).
Another example of this eccentric self-indulgence, is in the scene directly before, where Thor states that he's going back to Asgard: Grandmaster notes "Ass-gard", chuckles to himself, and cuts the conversation off abruptly to play the keyboard.
That said, I think other people's explanation about dodging his real age question is equally valid - and I had not realised this interpretation myself, since I saw the Grandmaster's trailing-off less as a question, and more as part of the overarching comedy of the movie.
This trailing-off is my favourite scene from the movie - followed by the Wonka chair nervous breakdown, and scenes involving Korg - and I have rewatched it countless times. It's an example of offbeat humour, which is sorely missing in mainstream blockbusters today.
The movie has a wide scope of humour, from off-beat (e.g. Grandmaster), deadpan (e.g. Topaz, Korg), meta (e.g. the theatre production, Korg about Thor's hammer), immature (e.g. "Get help"), to foil/straight man double acts (e.g. Topaz vs Valkyrie, Hulk vs Thor, Grandmaster vs everyone), and it's a joy looking for all of them during rewatches. I look forward to seeing more of Waititi and Goldblum's comedic genius.