The thaumaturgy spell states
You manifest a minor wonder, a sign of supernatural power, within range. You create one of the following magical effects within range:
- Your voice booms up to three times as loud as normal for 1 minute.
- You cause flames to flicker, brighten, dim, or change color for 1 minute.
- You cause harmless tremors in the ground for 1 minute.
- You create an instantaneous sound that originates from a point of your choice within range, such as a rumble of thunder, the cry of a raven, or ominous whispers.
- You instantaneously cause an unlocked door or window to fly open or slam shut.
- You alter the appearance of your eyes for 1 minute.
If you cast this spell multiple times, you can have up to three of its 1-minute effects active at a time, and you can dismiss such an effect as an action.
The Sorcerer's Extended Spell Metamagic feature states:
When you cast a spell that has a duration of 1 minute or longer, you can spend 1 sorcery point to double its duration, to a maximum duration of 24 hours.
Even though the duration of thaumaturgy is 1 minute, the descriptions describes the effects as only lasting for 1 minute instead of "for the spell's duration". Can you use the Extended Spell option on the thaumaturgy spell to meaningful effect?
There are other situations with similar interactions such as using the Distant Spell Metamagic Option on the counterspell spell, as discussed in the question: "Can you effectively apply the Distant Spell metamagic option to a Counterspell?" Where the reaction needing to be within 60 feet of you isn't overridden by the spell.
Best Answer
Technically no, but if this becomes an issue at your table, something, somewhere, has gone horribly wrong.
Following the logic of the linked answer to the adjusted range of Counterspell, a literal reading would suggest you can't extend the duration of the Thaumaturgy's effect beyond one minute because the effect duration is not literally linked to the spell's duration.
That is, it does not say "for the spell's duration [you can create an effect]."
But this is a very strict literal interpretation of the rules. We're taking about augmenting a cantrip with limited class resources. As a DM, I'd just let the player trying this have it.