Been down this road many times... somehow I am the "go to guy" for info on cheap/budget Android devices. Maybe because I am... umm... frugal.
My current recommendations...
1) Axon 7 Mini for $199 - Amazon still stocks them, and if you're in the US many BestBuys still have local stock in stores. This is a fantastic device for the price (it MSRP's at $299 and was worth it at that), good battery life, amazing sound quality, AMOLED screen, etc. I still think this is an incredibly under-rated device, my son has one and loves it, and it gets updates WAY faster than any Moto I have had in the last 2 yerars. He got the Bluebourne fix mid-September and has been on 7.1.1 since spring.
2) Moto G5 Plus 64GB version - If it's on-sale, at MSRP it's overpriced (and this is my DD phone), at $249 or less it's a good deal, but skip the 2/32GB version because the stock OS was obviously designed around 4GB of RAM.
3) Honor 6x - Great budget device for $199, if you can deal with the EMUI interface. The Honor 8 is going for $249 on Amazon and is a step up from this device too. Personally, I haven't been a huge fan of Huawei since the Ascend Mate 2, just don't like EMUI, but there is no questioning they have stepped up their game in the last couple years, people like these phones, and they give good bang for the buck.
4) Blu R1 Plus - I don't care what people say about Blu, this is a solid device with great battery life and a bargain at $160 from Amazon or BestBuy. 3GB RAM, 32GB storage, microSD support, 4000mAh battery, almost pure Android, and comes with a free case. Skip the 2GB RAM or 16GB storage variants though, for the $20 savings it's not worth it.
EDIT: I forgot one, often my recommendation in an "emergency" when you don't have a lot of cash but need a decent phone NOW.
E911) BLU Studio XL2 - Every BestBuy I have been in (MN, SD, ND, IA, WI, NE) stocks this phone so I assume it is available almost everywhere there is a BB... $129 and has a 2GB RAM, 16GB storage, 6" display, 4900mAh battery, dual SIM and microSD card support, 13MP camera, almost pure Android, and includes a case. It is no flagship but it is solid and will run for 2 days on a single charge for most people, 4 days if you keep the usage light.
Other notable mentions would be the Moto G4 Plus (a bargain if you can still find them), LG G5 (getting down around $249 for a 2 year flagship, not bad), Nokia 6...
Beyond those recommendations, we would have to nail down exactly what "budget" means to you and what you need... And if your on Verizon or Sprint, that changes things a bit too. There are no phones that retail below $120 I am recommending to anyone right now.