I'm a little rusty on this time period but I'll give it a shot. Disclaimer, there is more guessing here than anything else...so...if anyone sees something that certainly won't happen because of XYZ...feel free to point it out because I am sure there are a few things here that just won't happen...I just don't know which.
A good POD would be to prevent Trajedo's reelection in 1876. No 1876 reelection means that Diaz doesn't launch another revolt. It could mean that Jose Maria Iglesias would become President 1876-1880 instead of taking power after nullifying Trajedo's reelection making his presidency unquestionably legitimate. This means no Diaz interim presidency after demanding an election in light of his revolt that helped oust Trajedo.
Unless Diaz is willing to launch an unfounded military revolt, this knocks him out of guaranteed electoral victory for 1880 since the 1880-1884. But let's assume that he still manages to eek out a victory, Manuel Gonzalez IOTL was handpicked by Diaz to run in 1880 since Diaz would violate his own no-reelection cause. Manuel Gonzalez was pretty much a political clone of Diaz so we can assume a similar administration. Diaz was just more self interested, pragmatic, and harsh. Manuel Gonzalez made a few tough calls that got him some level of unpopularity including recognizing Mexican debt to the UK in an attempt to make nice with European powers. This lead to riots in Mexico city which were harshly suppressed. He left as a very unpopular figure.
ITTL we can assume that Diaz would do many of the positive things. Spurr renewed industrialization, expand the railroads and highways (read "roads") and help establish an effective telegraph network as well as a submarine cable and invite foreign investment. Then the negative things. Deficit spending, foreign investment, a new debt to bay (to the UK) harshly suppressing protests that turn to riots or exacerbating protests that then as a result become riots.
IOTL he was seen as the man who could hold the country together, which won him a second term in office (justified by it not being a consecutive term). I imagine Iglesias or someone else might support some opposition to Diaz's boy, Manuel gonzalez. If Mexico is lucky, maybe even have a birth of political parties out of the 1884 election. The Autocratic pro-Diaz party and the Democratic pro-Iglesias party. Let's call the democratic oriented party the "Federalist Liberal Party" and the autocratic oriented party the "Nationalist Liberal Party" Obviously these would not be the names in reality but I am throwing some guesses so bare with me. Manuel Gonzalez OTL was...not popular so maybe Diaz's performance leaves him not as popular as he wanted to be (or was OTL). Manuel Gomez of the Nationalist Liberal Party (Partido Nacionalista Liberal, PNL) wins, Diaz's policies continues and people get angry and flee towards the Federalist Liberal Party. 1884-1888 is 4 years without Diaz, Diaz losses his bid in 1888 giving a new president from the Federalist Liberal Party (Partido Federalista Liberal, PFL).
Assuming that this opposition party would win showcasing Diaz's faults, they'd then operate in areas Diaz neglected...meaning helping the people of Mexico (A revolutionary concept for Mexico...hints the OTL revolution). That means land grants, education reform, stopping further foreign investors (not that that was actually that bad of a thing but perception is key) and maybe renegotiating foreign debts? So Mexico is now a somewhat polarized liberal democracy that would have problems of banditry, corruption, economic inequality, and a sluggish economy with government debt and deficit spending.
Other problems are an appallingly high illiteracy rate (Hovering somewhere between 80%-90%, low immigration (liberals wanted European immigration so they saw this as a problem). Then there's the issue of religious freedom that was left out of the 1857 constitution. A resurgence of Catholic conservatives without an all powerful Diaz "dictatorship" managing the situation might give rise to a new conservative party, unless Diaz co-opts it to sure up support. Now the PFL president from 188-1892 could try to amend the constitution to make it clear that no reelection mean no second term at all, not just "no consecutive terms". If so, that effectively ends Diaz's bid for the presidency forcing him to lead from behind the scenes as president of the PNL.
Edit: Of course he can always catch a stray bullet in a failed Plan of Tuxtepec or simply lose and get executed for treason. But this opens a door for a "Tejadato"...or a Gonzalezato....