Will you recommend an activity for my 5-year-old?

Taekwondo
Kids love that stuff.
Or Lacrosse

I’d say baseball or basketball. You could play catch with him and have some nice moments. Basketball he can practice on his own. How’s his hand eye coordination?

Are you in the U.S.? Soccer and tee ball season will be here soon. They usually have at least one practice and one game per week. It will keep your child active and he’ll learn how to work with a team.

What does he like to do, @phaedryx? Is he athletic? Does he like running or reading? Is he an inside kid or an outside kid?

Kiddie gym. it burns energy.

Theater, soccer, dance, swimming, track, cub scouts, Campfire USA

Check out personality profiling through Meyers-Briggs or Enneagram to see if you can determine his natural brain functions and orientations for a birds eye view of what will be motivating factors for his interests and ways he identifies with his environment.
Regardless of personality style I am huge advocate for teaching children a love and curiosity of nature, the natural world and the arts.

Load him up with art supplies, animal and nature books, films, age appropriate science projects. Personality theory is huge in determining areas of excellence with children as we are all born with innate tendencies towards how we explore and assimilate our environments.
Many NTP personalities are often seen as ADD or ADHD when, infact, they are gifted children with special needs for maximum learning motivation.
5 years old is still very young to expect a long attention span, but I strongly suggest an attempt at personality profiling to glean as much info. as you can towards providing for his individual needs and natural talents. :-)

I agree with @josie‘s recommendation as long as it is a good dojo with an excellent teacher.

I also love the idea of a sports team. At five most of them are more for fun than competition and they help teach team work, good sportsmanship, and gross motor skills.

If you aren’t ready to commit for months at a time, check out your local library to see what special events they have. Ours has Lego building days, art classes, and nature walks! Bonus: All for free!

At that age (during winter and bad weather) my sisters and I were riding tricycles around the cellar on a track that my dad had made for us (with a tiny little “hill” made of plywood over wedges). In the better weather we’d be riding them outside, or sledding, roller skating, hide and seek, tag and other group activities.

swimming
bicycle

Having some time with you—whatever it is, will be a delight for him.

Dance. (Or (a) sport(s) or martial art.) Like said above, basically.
Maybe a dog, if he can handle a portion of the responsibility.

Swim club pollywog division.

Art. Anything to do with art. I am not sure about your city, but where I live the library holds free art sessions for children a few times a week.

People have said most of what my suggestions would be.

Dance
Swimming
Soccer
Gymnastics
Tennis
Martial Arts

Does he like team sports? Or, prefer individual athletics?

I especially love dance and martial arts for how it helps develop a mind body connection. Awareness of where our body is in space and how to control it. It helps later with all sorts of coordination, posture, and confidence.

foreign language, Martial Arts

Let your child play a sport!!