Let's say Bobby has a shield in his left hand and a whip in his right hand.
Bobby also has a couple of nets, and is mounted on his favorite pony.
It's Bobby's turn, and he sees an enemy within reach of his nets. He drops his whip [no action] to grab a net [free object interaction] before throwing it.
But Bobby is concerned: he doesn't want to lose his precious whip because he dropped it on the ground far away, so he would like to have his weapon somehow roped to either himself or his pony so that it will stay dangling close to him when he needs it again.
Can Bobby do that (attach a weapon to himself/his mount)? Would there be drawbacks (attack disadvantage/penalty with the weapon due to its attached status; attack disadvantage/penalty for the other weapon due to the attached one being in the way; possibility for the dangling weapon to hurt him and/or the mount; etc.)?
Best Answer
No, there are no rules for 5e that cover this situation.
As you stated, the Rules in the PHB on page 190 for interacting with an object let you pick one up at no cost (pending no other interactions during your turn) are sufficient for characters on the ground, mounted characters still have a problem with reaching their item.
Caveat Emptor on a homebrew solution
Part of the current ruleset also allows you to pick up someone else's item that is on the ground. The Disarming Maneuver from the fighter would be rendered useless should a weapon cord be made available.
However, should you want to introduce the weapon cord, there is an excellent suggestion for homebrewing in AS.Beauliu's answer.