[4] On March 20, 2015, it was announced that Rodríguez had been sent on loan to the Dynamo's USL affiliate club Charleston Battery.

Rodriguez would score his first goal for RGVFC on May 21, finding the back of the net twice in a 4–0 win over Swope Park Rangers.

[14] He finished the regular season with 9 appearances, 2 goals, and 1 assist as he helped his hometown team make it back to the MLS playoffs for the first time since 2013.

[19] The Dynamo went on to win the Open Cup, the first in club history, with Rodriguez playing in every game and getting 3 goals and an assist.

[20] In the Dynamo's next match, Memo got the start against Montreal and scored on a shot from 22 yards out, helping Houston to a 2–1 win.

[2] After appearing in a 1–1 draw with Sporting Kansas City on June 1, he missed around 2 months due to a hamstring injury.

[24] His return to the lineup was unable to save the season as the Dynamo missed out on the playoffs despite the best start in club history.

[25] Rodriguez was named as the Dynamo Young Player of the Year for his strong season, scoring 7 and assisting 5 in 26 MLS appearances.

In Houston's first game back, Rodriguez scored twice in a 3–3 draw with LAFC on July 13, a performance that saw him named to the MLS Team of the Week.

[33] On April 16, 2021, in the opening match of the season, Rodriguez scored once to help the Dynamo defeat the San Jose Earthquakes 2–1.

[30] Rodriguez made his first appearance of the 2022 season on February 27, getting the start in a 0–0 draw vs Real Salt Lake in matchweek 1.

[37] He picked up his first assist of the season on March 19, setting up Tyler Pasher in the 90th minute to draw 1–1 with the Colorado Rapids.

[38] On January 5, 2023, Rodríguez signed with LA Galaxy as a free agent on a one-year deal with club options for 2024 and 2025.

[3][28] He is eligible to represent both the United States and Mexico at the international level and has expressed a willingness to play for either nation.