So both those are reasonable if you read the meta right. I would also say that depending on what budget means (0-proxy vs 5-10 proxy vs infinite proxy) and also how much money is budget for you ($400 vs $1000 vs anything but power/shops/bazaars) makes a big difference in answers to this question.
In general I would tend toward unpowered blue deck over something like red prison in vintage? In a 5-proxy meta I would do this every time (3 mox/lotus 1 time walk 1 ancestral, and the rest of the deck is legacy cards).
Budget decks are all going to be somewhat limited in power level. You can, however, key in on a big matchup (or two) and get a service-able deck that can go on a run when it hits its matchups. With all the recent printings, I’d look long and hard at GW hatebears. You can cover a lot of ground with access to 12 Null Rods, and various GY/storm hate options.
In this case, "budget" means she can play it in Vintage Champs and qualify for the budget prize.
I do like G/W Hate Bears. Now I need to find a mostly updated list.
There’s probably a no Bazaar/power Survival build. I’d probably start BUG to use Deathrite and Bazaar Trademage, but a GRx build using Faithless Looting might also do well now that Mental Misstep is restricted.
Another option would be Vintage Freebird. @IAmActuallyLvl1 has been doing an incredible amount of work with this deck that started as a powerful Legacy deck. The Legacy version almost had an undefeated run in the Legacy Challenge, losing in the finals.
The powered version of this deck runs Black Lotus, Mox Jet, and Mox Ruby. I've replaced them with the two extra lands that were in the Legacy build and the third Bedlam Reveler. One benefit of this deck for EW is that the maindeck is only two cards away from being Legacy legal!
Finally, don't play G/W Hatebears. Playing unpowered doesn't mean you should play underpowered. There are still plenty of viable, strong decks that exist without the artifact mana, especially in a post-Mental Misstep world.
Vintage Freebird (Unpowered)
Any Fastbond deck that is looking to cast Life from the Loam is going to feel underpowered without access to Bazaar of Baghdad. Still, decks like Turbo/Slow/Hogaak Depths and AmuLIT offer those that are looking to cast Fastbond reasonable shells that don't lose a core strategy to their budget.
@Brass-Man also had a really cool BG survival deck that could very easily be made powerless. It also had very few (if any) duals.
Moxen in that deck aren't really great, -you play Collector Ouphe which shuts them off. They also can't be tutored with survival and Elvish Spirit Guide can be.