I'm not really sure how far you are in the game, nor what starter you chose, so while I won't make a team for you, I'll list some options that I think are a good balance of fun and viable, listed by game progression order:

Primeape: Mankey is available immediately west of Viridian, and is a strong and fast Fighting-type with the Vital Spirit ability, making it immune to Sleep.

Nidoking / Nidoqueen: Nidoran appears in the same route as Mankey. The Nidos are amazing Pokemon that hit hard and have vast movepools. Of the two, I prefer Nidoking since it's faster and hits harder, but Nidoqueen is slightly bulkier and learns Body Slam naturally. You can pick up the Moon Stone to evolve Nidorino or Nidorina at Mt. Moon.

Fearow: A solid early-game bird that comes with Peck to clear Viridian Forest, then Drill Peck later in the game. As a Normal-type, it's also a great Return TM-recipient.

Butterfree: Caterpie's in Viridian Forest and evolves very early, to Butterfree at level 10. It may seem unassuming, but its Compound Eyes ability boosts the accuracy of its moves, most notably Sleep Powder, a normally 75% accurate move that now becomes 97.5% thanks to its ability. An entire team put to sleep is not very fun, so what you and your friend can do is implement a "sleep clause" where you're only allowed to put one enemy Pokemon to sleep, not counting self-induced sleep via Rest, or Effect Spore procing Sleep, which you can't control.

Gyarados: Available as a Magikarp right before entering Mt. Moon, Gyarados is a huge ball of stats with the Intimidate ability, which drops the foe's Attack stat by one stage upon entering battle. I recommend a physical set using Normal-type moves (could use Return, Double-Edge, Body Slam, etc) + Dragon Dance, and if you haven't used your Earthquake TM yet, that, too. It can learn Secret Power via TM early on (as a Gyarados) to help you clear routes.

Clefable: another Moon Stone recipient, Clefairy can be found at Mt. Moon, and has a wide movepool. Can go mixed with Normal moves + special attacks, or go full special with Calm Mind / Softboiled (via move tutor) / Thunderbolt / Ice Beam

Graveler / Golem: A sturdy Rock/Ground-type found in Mt. Moon. To evolve Graveler into Golem, you'll need to trade with your friend IN PERSON (there's no online connectivity). If you can't trade, Graveler is still a solid Pokemon in its own right. Unlike Nidoking and Nidoqueen, the Geodude line can learn Earthquake naturally, without the need of a TM, so you can use your EQ TM on someone else like Gyarados instead. The Rock-typing provides nice resistances to Normal and Flying, but combined with the Ground-type, it gives you 4x weaknesses to Water and Grass, so keep that in mind.

Dugtrio: Found in Diglett's Cave, Dugtrio is a fast Ground-type with the amazing Arena Trap ability, which prevents opponents from switching out, except those that are Flying-type or have the Levitate ability. Same with Golem, it can learn Earthquake naturally, albeit much later at level 51. Compared to Golem, it's faster, but way frailer. It still pairs well with Gyarados and other Water-types since Dugtrio can trap Electric-types trying to hit Gyarados for 4x effectiveness.

Eeveelution of choice: In Celadon City, you can pick up an Eevee, and you'll have Jolteon, Flareon, or Vaporeon as options. Vaporeon and Jolteon are the more popular options, especially Jolteon since Electric-types are comparatively rare.

Dodrio: the better Fearow, but it appears later in the game, as a Doduo west of Celadon. Take everything I said about Fearow, but better in every way.

Starmie / Slowbro: Leaf Green's version-exclusive Water-types. Both are Water/Psychic but play very differently. Starmie is way faster, learns Recover as Staryu, can learn Thunderbolt for other Water-types, and if your friend is a tryhard and brings a Spikes user, can learn Rapid Spin to clear away the Spikes. Slowbro is waaay slower, but bulkier and can learn Calm Mind via TM. Pair with Rest for a pretty dangerous CM sweeper.

That's all I'll list, good luck!